Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case.) In parts (a)-(f), v 1 , ..., v p are vectors in a nonzero finite-dimensional vector space V , and S = { v 1 , ..., v p }. a. The set of all linear combinations of v 1 , ..., v p is a vector space. b. If { v 1 , ..., v p−1 } spans V , then S spans V . c. If { v 1 , ..., v p−1 } is linearly independent, then so is S . d. If S is linearly independent, then S is a basis for V . e. If Span S = V , then some subset of S is a basis for V . f. If dim V = p and Span, S = V , then S cannot be linearly dependent. g. A plane in ℝ 3 is a two-dimensional subspace. h. The nonpivot columns of a matrix are always linearly dependent. i. Row operations on a matrix A can change the linear dependence relations among the rows of A . j. Row operations on a matrix can change the null space. k. The rank of a matrix equals the number of nonzero rows. l. If an m × n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of A x = 0 is m − k . m. If B is obtained from a matrix A by several elementary row operations, then rank B = rank A . n. The nonzero rows of a matrix A form a basis for Row A . ○. If matrices A and B have the same reduced echelon form, then Row A = Row B . p. If H is a subspace of ℝ 3 , then there is a 3 × 3 matrix A such that H = Col A . q. If A is m × n and rank A = m , then the linear transformation x ↦ A x is one-to-one. r. If A is m × n and the linear transformation x ↦ Ax is onto, then rank A = m . s. A change-of-coordinates matrix is always invertible. t. If B = { b 1 , ..., b n } and C = { c 1 , ..., c n } are bases for a vector space V , then the j th column of the change-of-coordinates matrix is the coordinate vector [c j ] B .

Question

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Answer 1

Textbook Question

Chapter 4, Problem 1SE

Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case.) In parts (a)-(f), v₁, ..., v_p are vectors in a nonzero finite-dimensional vector space V, and S = {v₁, ..., v_p}.

a. The set of all linear combinations of v₁, ..., v_p is a vector space.
b. If {v₁, ..., v_p−1} spans V, then S spans V.
c. If {v₁, ..., v_p−1} is linearly independent, then so is S.
d. If S is linearly independent, then S is a basis for V.
e. If Span S = V, then some subset of S is a basis for V.
f. If dim V = p and Span, S = V, then S cannot be linearly dependent.
g. A plane in ℝ³ is a two-dimensional subspace.
h. The nonpivot columns of a matrix are always linearly dependent.
i. Row operations on a matrix A can change the linear dependence relations among the rows of A.
j. Row operations on a matrix can change the null space.
k. The rank of a matrix equals the number of nonzero rows.
l. If an m × n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax = 0 is m − k.
m. If B is obtained from a matrix A by several elementary row operations, then rank B = rank A.
n. The nonzero rows of a matrix A form a basis for Row A.
○. If matrices A and B have the same reduced echelon form, then Row A = Row B.
p. If H is a subspace of ℝ³, then there is a 3 × 3 matrix A such that H = Col A.
q. If A is m × n and rank A = m, then the linear transformation x ↦ Ax is one-to-one.
r. If A is m × n and the linear transformation x ↦ Ax is onto, then rank A = m.
s. A change-of-coordinates matrix is always invertible.
t. If B = {b₁, ..., b_n} and C = {c₁, ..., c_n} are bases for a vector space V, then the jth column of the change-of-coordinates matrix is the coordinate vector [c_j]_B.

a.

Expert Solution

To determine

To find: Whether the statement “The set of all linear combinations of v1,…,vp is a vector space” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here, the given vectors are v1,…,vp .

The span { v1,…,vp } always form a vector subspace of the vector space.

Thus, the linear combinations of v1,…,vp are also a vector space.

Hence, the statement is true.

b.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} spans V, then S spans V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The set S is S={v1,…,vp} .

It is given that the set {v1,…,vp−1} spans V. That is, span {v1,…,vp−1} = V .

That is, every element in the vector space V can be expressed by using the vectors {v1,…,vp−1} .

Here, the set S={v1,…,vp} , the set contain more than one element than the set {v1,…,vp−1} .

The smaller set {v1,…,vp−1} spans V which implies the bigger set S={v1,…,vp} also spans V.

If the vector space V is spanned by {v1,…,vp−1} , then its linear combinations also span V.

Hence, the statement is true.

c.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} is linearly independent, then so is S” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If {v1,…,vp−1} is a set of linearly independent vectors, then it does not imply that the set {v1,…,vp−1,vp} is also linearly independent as vp may be expressed as a linear combination of {v1,…,vp−1} .

Therefore, it does not imply that S is linearly independent.

Hence, the statement is false.

d.

Expert Solution

To determine

To find: Whether the statement “If S is linearly independent then S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

A set of vectors {v1,…,vp} is a basis for a vector space V if the following two conditions are satisfied:

1. The set of vectors {v1,…,vp} is a linearly independent set.

2. The set {v1,…,vp} spans the vector space V.

It can be seen that the second condition is not satisfied.

Hence, the statement is false.

e.

Expert Solution

To determine

To find: Whether the statement “If Span S=V , then some subset of S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that the vector space V is spanned by S, which is nonzero set.

Suppose the set S is linearly independent then, the set S form a basis for V.

Suppose the set S is linearly dependent then, some subset of S linearly independent and which spans V.

That is, some subset of S form a basis for V.

Hence, the statement is true.

f.

Expert Solution

To determine

To find: Whether the statement “If dimV=p and Span S=V , then S cannot be linearly dependent.” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that dimV=p . That is, the basis of the vector space contain p elements

Here, Span S=V , that is the set contain p elements span the p dimensional space.

Which implies the p elements is linearly independent.

Hence, the statement is false.

g.

Expert Solution

To determine

To find: Whether the statement “A plane in ℝ3 is a two-dimensional subspace” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Every plane in ℝ3 to be a two dimensional subspace is false.

Sometimes plane in ℝ3 does not form a vector space. Only the plane passing through origin form a subspace of ℝ3 .

Hence, the statement is false.

h.

Expert Solution

To determine

To find: Whether the statement “The non-pivot columns of a matrix are always linearly dependent” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Consider the matrix [1000001000000000] .

Here non-pivot columns are linearly independent.

Hence, the statement is false.

i.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix A can change the linear dependence relations among the rows of A” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Row operations on matrix A change the matrix. Thus, this changes the dependence relation among the rows of a matrix.

Hence, the statement is true.

j.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix can change the null space” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Row operations do not change the solution set of the system Ax=0 .

Therefore, row operations do not change the null space.

Hence, the statement is false.

k.

Expert Solution

To determine

To find: Whether the statement “The rank of a matrix equals the number of nonzero rows” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The rank of a matrix A is the dimension of the column space of A.

The dimension of column space of A is the number of pivot columns in A.

Therefore, the rank of matrix equals the number of pivot columns.

Consider the matrix [1111] .

The above matrix has 2 rows but rank of the matrix is 1.

Hence, the statement is false.

l.

Expert Solution

To determine

To find: Whether the statement “If an m×n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax=0 is m−k ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If U has k nonzero rows then, rank A=k .

According to the Rank Theorem, the rank of an m×n matrix A is same as the dimension of the column space of A and is equal to the number of pivot positions in A such that,

rank A+dimNul A=ndimNul A=n−rank A=n−k

Hence, the statement is false.

m.

Expert Solution

To determine

To find: Whether the statement “If B is obtained from a matrix A by several elementary row operations, then rank A=rank B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Elementary row operations does not change the number of pivot columns and hence, does not change the rank of a matrix.

Therefore, the rank of matrix B will be same as the rank of matrix A.

Hence, the statement is true.

n.

Expert Solution

To determine

To find: Whether the statement “The nonzero rows of a matrix A form a basis for Row A” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

To form a basis for A, the rows have to span A and should also be linearly independent.

The nonzero rows of a matrix A span Row A but that does not guarantee that they are linearly independent.

Hence, the statement is false.

o.

Expert Solution

To determine

To find: Whether the statement “If matrices A and B have the same reduced echelon form, then Row A=Row B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The nonzero rows of the echelon form of a matrix, form a basis for the row space of that matrix.

If the echelon form for two matrices is same, then the basis for the row spaces is also same.

Since, row spaces are vector spaces and if two vector spaces have same basis, then the vector spaces are same.

Hence, the statement is true.

p.

Expert Solution

To determine

To find: Whether the statement “If H is a subspace of ℝ3 , then there is a 3×3 matrix A such that H=Col A ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

If H is a zero, 1, 2, or 3 dimensional subspace of ℝ3 , there will always be a corresponding 3×3 matrix A formed by the basis of H.

The basis of H will then be in the column space of A.

Therefore, H=Col A .

Hence, the statement is true.

q.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and rank A=m , then the linear transformation x↦Ax is one-to-one” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

The transformation x↦Ax is one-to-one if and only if null space of A is zero.

Here, rank of the matrix is A thus, by the rank nullity theorem null space of A contain n−m elements.

Hence, the statement is false.

r.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and the linear transformation x↦Ax is onto, then rank A=m .)” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

If the transformation is onto then, Col A=ℝm

The rank of a matrix A is the dimension of the column space of A.

Therefore, the rank of A is m.

Hence, the statement is true.

s.

Expert Solution

To determine

To find: Whether the statement “A change-of-coordinate matrix is always invertible” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set.

The matrix PC←B is a square matrix thus; it must be an invertible matrix.

Hence, the statement is true.

t.

Expert Solution

To determine

To find: Whether the statement “If B={b1,…,bn} and C={c1,…,cn} are bases for a vector space V, then the j th column of the change-of-coordinates matrix PC←B is the coordinate vector [cj]B ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The jth column of the change-of-coordinates matrix PC←B is [bj]C .

Hence, the statement is false.

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Answer 2

Textbook Question

Chapter 4, Problem 1SE

Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case.) In parts (a)-(f), v₁, ..., v_p are vectors in a nonzero finite-dimensional vector space V, and S = {v₁, ..., v_p}.

a. The set of all linear combinations of v₁, ..., v_p is a vector space.
b. If {v₁, ..., v_p−1} spans V, then S spans V.
c. If {v₁, ..., v_p−1} is linearly independent, then so is S.
d. If S is linearly independent, then S is a basis for V.
e. If Span S = V, then some subset of S is a basis for V.
f. If dim V = p and Span, S = V, then S cannot be linearly dependent.
g. A plane in ℝ³ is a two-dimensional subspace.
h. The nonpivot columns of a matrix are always linearly dependent.
i. Row operations on a matrix A can change the linear dependence relations among the rows of A.
j. Row operations on a matrix can change the null space.
k. The rank of a matrix equals the number of nonzero rows.
l. If an m × n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax = 0 is m − k.
m. If B is obtained from a matrix A by several elementary row operations, then rank B = rank A.
n. The nonzero rows of a matrix A form a basis for Row A.
○. If matrices A and B have the same reduced echelon form, then Row A = Row B.
p. If H is a subspace of ℝ³, then there is a 3 × 3 matrix A such that H = Col A.
q. If A is m × n and rank A = m, then the linear transformation x ↦ Ax is one-to-one.
r. If A is m × n and the linear transformation x ↦ Ax is onto, then rank A = m.
s. A change-of-coordinates matrix is always invertible.
t. If B = {b₁, ..., b_n} and C = {c₁, ..., c_n} are bases for a vector space V, then the jth column of the change-of-coordinates matrix is the coordinate vector [c_j]_B.

a.

Expert Solution

To determine

To find: Whether the statement “The set of all linear combinations of v1,…,vp is a vector space” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here, the given vectors are v1,…,vp .

The span { v1,…,vp } always form a vector subspace of the vector space.

Thus, the linear combinations of v1,…,vp are also a vector space.

Hence, the statement is true.

b.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} spans V, then S spans V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The set S is S={v1,…,vp} .

It is given that the set {v1,…,vp−1} spans V. That is, span {v1,…,vp−1} = V .

That is, every element in the vector space V can be expressed by using the vectors {v1,…,vp−1} .

Here, the set S={v1,…,vp} , the set contain more than one element than the set {v1,…,vp−1} .

The smaller set {v1,…,vp−1} spans V which implies the bigger set S={v1,…,vp} also spans V.

If the vector space V is spanned by {v1,…,vp−1} , then its linear combinations also span V.

Hence, the statement is true.

c.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} is linearly independent, then so is S” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If {v1,…,vp−1} is a set of linearly independent vectors, then it does not imply that the set {v1,…,vp−1,vp} is also linearly independent as vp may be expressed as a linear combination of {v1,…,vp−1} .

Therefore, it does not imply that S is linearly independent.

Hence, the statement is false.

d.

Expert Solution

To determine

To find: Whether the statement “If S is linearly independent then S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

A set of vectors {v1,…,vp} is a basis for a vector space V if the following two conditions are satisfied:

1. The set of vectors {v1,…,vp} is a linearly independent set.

2. The set {v1,…,vp} spans the vector space V.

It can be seen that the second condition is not satisfied.

Hence, the statement is false.

e.

Expert Solution

To determine

To find: Whether the statement “If Span S=V , then some subset of S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that the vector space V is spanned by S, which is nonzero set.

Suppose the set S is linearly independent then, the set S form a basis for V.

Suppose the set S is linearly dependent then, some subset of S linearly independent and which spans V.

That is, some subset of S form a basis for V.

Hence, the statement is true.

f.

Expert Solution

To determine

To find: Whether the statement “If dimV=p and Span S=V , then S cannot be linearly dependent.” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that dimV=p . That is, the basis of the vector space contain p elements

Here, Span S=V , that is the set contain p elements span the p dimensional space.

Which implies the p elements is linearly independent.

Hence, the statement is false.

g.

Expert Solution

To determine

To find: Whether the statement “A plane in ℝ3 is a two-dimensional subspace” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Every plane in ℝ3 to be a two dimensional subspace is false.

Sometimes plane in ℝ3 does not form a vector space. Only the plane passing through origin form a subspace of ℝ3 .

Hence, the statement is false.

h.

Expert Solution

To determine

To find: Whether the statement “The non-pivot columns of a matrix are always linearly dependent” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Consider the matrix [1000001000000000] .

Here non-pivot columns are linearly independent.

Hence, the statement is false.

i.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix A can change the linear dependence relations among the rows of A” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Row operations on matrix A change the matrix. Thus, this changes the dependence relation among the rows of a matrix.

Hence, the statement is true.

j.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix can change the null space” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Row operations do not change the solution set of the system Ax=0 .

Therefore, row operations do not change the null space.

Hence, the statement is false.

k.

Expert Solution

To determine

To find: Whether the statement “The rank of a matrix equals the number of nonzero rows” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The rank of a matrix A is the dimension of the column space of A.

The dimension of column space of A is the number of pivot columns in A.

Therefore, the rank of matrix equals the number of pivot columns.

Consider the matrix [1111] .

The above matrix has 2 rows but rank of the matrix is 1.

Hence, the statement is false.

l.

Expert Solution

To determine

To find: Whether the statement “If an m×n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax=0 is m−k ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If U has k nonzero rows then, rank A=k .

According to the Rank Theorem, the rank of an m×n matrix A is same as the dimension of the column space of A and is equal to the number of pivot positions in A such that,

rank A+dimNul A=ndimNul A=n−rank A=n−k

Hence, the statement is false.

m.

Expert Solution

To determine

To find: Whether the statement “If B is obtained from a matrix A by several elementary row operations, then rank A=rank B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Elementary row operations does not change the number of pivot columns and hence, does not change the rank of a matrix.

Therefore, the rank of matrix B will be same as the rank of matrix A.

Hence, the statement is true.

n.

Expert Solution

To determine

To find: Whether the statement “The nonzero rows of a matrix A form a basis for Row A” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

To form a basis for A, the rows have to span A and should also be linearly independent.

The nonzero rows of a matrix A span Row A but that does not guarantee that they are linearly independent.

Hence, the statement is false.

o.

Expert Solution

To determine

To find: Whether the statement “If matrices A and B have the same reduced echelon form, then Row A=Row B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The nonzero rows of the echelon form of a matrix, form a basis for the row space of that matrix.

If the echelon form for two matrices is same, then the basis for the row spaces is also same.

Since, row spaces are vector spaces and if two vector spaces have same basis, then the vector spaces are same.

Hence, the statement is true.

p.

Expert Solution

To determine

To find: Whether the statement “If H is a subspace of ℝ3 , then there is a 3×3 matrix A such that H=Col A ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

If H is a zero, 1, 2, or 3 dimensional subspace of ℝ3 , there will always be a corresponding 3×3 matrix A formed by the basis of H.

The basis of H will then be in the column space of A.

Therefore, H=Col A .

Hence, the statement is true.

q.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and rank A=m , then the linear transformation x↦Ax is one-to-one” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

The transformation x↦Ax is one-to-one if and only if null space of A is zero.

Here, rank of the matrix is A thus, by the rank nullity theorem null space of A contain n−m elements.

Hence, the statement is false.

r.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and the linear transformation x↦Ax is onto, then rank A=m .)” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

If the transformation is onto then, Col A=ℝm

The rank of a matrix A is the dimension of the column space of A.

Therefore, the rank of A is m.

Hence, the statement is true.

s.

Expert Solution

To determine

To find: Whether the statement “A change-of-coordinate matrix is always invertible” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set.

The matrix PC←B is a square matrix thus; it must be an invertible matrix.

Hence, the statement is true.

t.

Expert Solution

To determine

To find: Whether the statement “If B={b1,…,bn} and C={c1,…,cn} are bases for a vector space V, then the j th column of the change-of-coordinates matrix PC←B is the coordinate vector [cj]B ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The jth column of the change-of-coordinates matrix PC←B is [bj]C .

Hence, the statement is false.

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Answer 3

Textbook Question

Chapter 4, Problem 1SE

Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case.) In parts (a)-(f), v₁, ..., v_p are vectors in a nonzero finite-dimensional vector space V, and S = {v₁, ..., v_p}.

a. The set of all linear combinations of v₁, ..., v_p is a vector space.
b. If {v₁, ..., v_p−1} spans V, then S spans V.
c. If {v₁, ..., v_p−1} is linearly independent, then so is S.
d. If S is linearly independent, then S is a basis for V.
e. If Span S = V, then some subset of S is a basis for V.
f. If dim V = p and Span, S = V, then S cannot be linearly dependent.
g. A plane in ℝ³ is a two-dimensional subspace.
h. The nonpivot columns of a matrix are always linearly dependent.
i. Row operations on a matrix A can change the linear dependence relations among the rows of A.
j. Row operations on a matrix can change the null space.
k. The rank of a matrix equals the number of nonzero rows.
l. If an m × n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax = 0 is m − k.
m. If B is obtained from a matrix A by several elementary row operations, then rank B = rank A.
n. The nonzero rows of a matrix A form a basis for Row A.
○. If matrices A and B have the same reduced echelon form, then Row A = Row B.
p. If H is a subspace of ℝ³, then there is a 3 × 3 matrix A such that H = Col A.
q. If A is m × n and rank A = m, then the linear transformation x ↦ Ax is one-to-one.
r. If A is m × n and the linear transformation x ↦ Ax is onto, then rank A = m.
s. A change-of-coordinates matrix is always invertible.
t. If B = {b₁, ..., b_n} and C = {c₁, ..., c_n} are bases for a vector space V, then the jth column of the change-of-coordinates matrix is the coordinate vector [c_j]_B.

a.

Expert Solution

To determine

To find: Whether the statement “The set of all linear combinations of v1,…,vp is a vector space” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here, the given vectors are v1,…,vp .

The span { v1,…,vp } always form a vector subspace of the vector space.

Thus, the linear combinations of v1,…,vp are also a vector space.

Hence, the statement is true.

b.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} spans V, then S spans V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The set S is S={v1,…,vp} .

It is given that the set {v1,…,vp−1} spans V. That is, span {v1,…,vp−1} = V .

That is, every element in the vector space V can be expressed by using the vectors {v1,…,vp−1} .

Here, the set S={v1,…,vp} , the set contain more than one element than the set {v1,…,vp−1} .

The smaller set {v1,…,vp−1} spans V which implies the bigger set S={v1,…,vp} also spans V.

If the vector space V is spanned by {v1,…,vp−1} , then its linear combinations also span V.

Hence, the statement is true.

c.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} is linearly independent, then so is S” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If {v1,…,vp−1} is a set of linearly independent vectors, then it does not imply that the set {v1,…,vp−1,vp} is also linearly independent as vp may be expressed as a linear combination of {v1,…,vp−1} .

Therefore, it does not imply that S is linearly independent.

Hence, the statement is false.

d.

Expert Solution

To determine

To find: Whether the statement “If S is linearly independent then S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

A set of vectors {v1,…,vp} is a basis for a vector space V if the following two conditions are satisfied:

1. The set of vectors {v1,…,vp} is a linearly independent set.

2. The set {v1,…,vp} spans the vector space V.

It can be seen that the second condition is not satisfied.

Hence, the statement is false.

e.

Expert Solution

To determine

To find: Whether the statement “If Span S=V , then some subset of S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that the vector space V is spanned by S, which is nonzero set.

Suppose the set S is linearly independent then, the set S form a basis for V.

Suppose the set S is linearly dependent then, some subset of S linearly independent and which spans V.

That is, some subset of S form a basis for V.

Hence, the statement is true.

f.

Expert Solution

To determine

To find: Whether the statement “If dimV=p and Span S=V , then S cannot be linearly dependent.” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that dimV=p . That is, the basis of the vector space contain p elements

Here, Span S=V , that is the set contain p elements span the p dimensional space.

Which implies the p elements is linearly independent.

Hence, the statement is false.

g.

Expert Solution

To determine

To find: Whether the statement “A plane in ℝ3 is a two-dimensional subspace” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Every plane in ℝ3 to be a two dimensional subspace is false.

Sometimes plane in ℝ3 does not form a vector space. Only the plane passing through origin form a subspace of ℝ3 .

Hence, the statement is false.

h.

Expert Solution

To determine

To find: Whether the statement “The non-pivot columns of a matrix are always linearly dependent” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Consider the matrix [1000001000000000] .

Here non-pivot columns are linearly independent.

Hence, the statement is false.

i.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix A can change the linear dependence relations among the rows of A” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Row operations on matrix A change the matrix. Thus, this changes the dependence relation among the rows of a matrix.

Hence, the statement is true.

j.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix can change the null space” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Row operations do not change the solution set of the system Ax=0 .

Therefore, row operations do not change the null space.

Hence, the statement is false.

k.

Expert Solution

To determine

To find: Whether the statement “The rank of a matrix equals the number of nonzero rows” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The rank of a matrix A is the dimension of the column space of A.

The dimension of column space of A is the number of pivot columns in A.

Therefore, the rank of matrix equals the number of pivot columns.

Consider the matrix [1111] .

The above matrix has 2 rows but rank of the matrix is 1.

Hence, the statement is false.

l.

Expert Solution

To determine

To find: Whether the statement “If an m×n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax=0 is m−k ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If U has k nonzero rows then, rank A=k .

According to the Rank Theorem, the rank of an m×n matrix A is same as the dimension of the column space of A and is equal to the number of pivot positions in A such that,

rank A+dimNul A=ndimNul A=n−rank A=n−k

Hence, the statement is false.

m.

Expert Solution

To determine

To find: Whether the statement “If B is obtained from a matrix A by several elementary row operations, then rank A=rank B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Elementary row operations does not change the number of pivot columns and hence, does not change the rank of a matrix.

Therefore, the rank of matrix B will be same as the rank of matrix A.

Hence, the statement is true.

n.

Expert Solution

To determine

To find: Whether the statement “The nonzero rows of a matrix A form a basis for Row A” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

To form a basis for A, the rows have to span A and should also be linearly independent.

The nonzero rows of a matrix A span Row A but that does not guarantee that they are linearly independent.

Hence, the statement is false.

o.

Expert Solution

To determine

To find: Whether the statement “If matrices A and B have the same reduced echelon form, then Row A=Row B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The nonzero rows of the echelon form of a matrix, form a basis for the row space of that matrix.

If the echelon form for two matrices is same, then the basis for the row spaces is also same.

Since, row spaces are vector spaces and if two vector spaces have same basis, then the vector spaces are same.

Hence, the statement is true.

p.

Expert Solution

To determine

To find: Whether the statement “If H is a subspace of ℝ3 , then there is a 3×3 matrix A such that H=Col A ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

If H is a zero, 1, 2, or 3 dimensional subspace of ℝ3 , there will always be a corresponding 3×3 matrix A formed by the basis of H.

The basis of H will then be in the column space of A.

Therefore, H=Col A .

Hence, the statement is true.

q.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and rank A=m , then the linear transformation x↦Ax is one-to-one” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

The transformation x↦Ax is one-to-one if and only if null space of A is zero.

Here, rank of the matrix is A thus, by the rank nullity theorem null space of A contain n−m elements.

Hence, the statement is false.

r.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and the linear transformation x↦Ax is onto, then rank A=m .)” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

If the transformation is onto then, Col A=ℝm

The rank of a matrix A is the dimension of the column space of A.

Therefore, the rank of A is m.

Hence, the statement is true.

s.

Expert Solution

To determine

To find: Whether the statement “A change-of-coordinate matrix is always invertible” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set.

The matrix PC←B is a square matrix thus; it must be an invertible matrix.

Hence, the statement is true.

t.

Expert Solution

To determine

To find: Whether the statement “If B={b1,…,bn} and C={c1,…,cn} are bases for a vector space V, then the j th column of the change-of-coordinates matrix PC←B is the coordinate vector [cj]B ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The jth column of the change-of-coordinates matrix PC←B is [bj]C .

Hence, the statement is false.

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Answer 4

Textbook Question

Chapter 4, Problem 1SE

Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case.) In parts (a)-(f), v₁, ..., v_p are vectors in a nonzero finite-dimensional vector space V, and S = {v₁, ..., v_p}.

a. The set of all linear combinations of v₁, ..., v_p is a vector space.
b. If {v₁, ..., v_p−1} spans V, then S spans V.
c. If {v₁, ..., v_p−1} is linearly independent, then so is S.
d. If S is linearly independent, then S is a basis for V.
e. If Span S = V, then some subset of S is a basis for V.
f. If dim V = p and Span, S = V, then S cannot be linearly dependent.
g. A plane in ℝ³ is a two-dimensional subspace.
h. The nonpivot columns of a matrix are always linearly dependent.
i. Row operations on a matrix A can change the linear dependence relations among the rows of A.
j. Row operations on a matrix can change the null space.
k. The rank of a matrix equals the number of nonzero rows.
l. If an m × n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax = 0 is m − k.
m. If B is obtained from a matrix A by several elementary row operations, then rank B = rank A.
n. The nonzero rows of a matrix A form a basis for Row A.
○. If matrices A and B have the same reduced echelon form, then Row A = Row B.
p. If H is a subspace of ℝ³, then there is a 3 × 3 matrix A such that H = Col A.
q. If A is m × n and rank A = m, then the linear transformation x ↦ Ax is one-to-one.
r. If A is m × n and the linear transformation x ↦ Ax is onto, then rank A = m.
s. A change-of-coordinates matrix is always invertible.
t. If B = {b₁, ..., b_n} and C = {c₁, ..., c_n} are bases for a vector space V, then the jth column of the change-of-coordinates matrix is the coordinate vector [c_j]_B.

a.

Expert Solution

To determine

To find: Whether the statement “The set of all linear combinations of v1,…,vp is a vector space” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here, the given vectors are v1,…,vp .

The span { v1,…,vp } always form a vector subspace of the vector space.

Thus, the linear combinations of v1,…,vp are also a vector space.

Hence, the statement is true.

b.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} spans V, then S spans V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The set S is S={v1,…,vp} .

It is given that the set {v1,…,vp−1} spans V. That is, span {v1,…,vp−1} = V .

That is, every element in the vector space V can be expressed by using the vectors {v1,…,vp−1} .

Here, the set S={v1,…,vp} , the set contain more than one element than the set {v1,…,vp−1} .

The smaller set {v1,…,vp−1} spans V which implies the bigger set S={v1,…,vp} also spans V.

If the vector space V is spanned by {v1,…,vp−1} , then its linear combinations also span V.

Hence, the statement is true.

c.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} is linearly independent, then so is S” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If {v1,…,vp−1} is a set of linearly independent vectors, then it does not imply that the set {v1,…,vp−1,vp} is also linearly independent as vp may be expressed as a linear combination of {v1,…,vp−1} .

Therefore, it does not imply that S is linearly independent.

Hence, the statement is false.

d.

Expert Solution

To determine

To find: Whether the statement “If S is linearly independent then S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

A set of vectors {v1,…,vp} is a basis for a vector space V if the following two conditions are satisfied:

1. The set of vectors {v1,…,vp} is a linearly independent set.

2. The set {v1,…,vp} spans the vector space V.

It can be seen that the second condition is not satisfied.

Hence, the statement is false.

e.

Expert Solution

To determine

To find: Whether the statement “If Span S=V , then some subset of S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that the vector space V is spanned by S, which is nonzero set.

Suppose the set S is linearly independent then, the set S form a basis for V.

Suppose the set S is linearly dependent then, some subset of S linearly independent and which spans V.

That is, some subset of S form a basis for V.

Hence, the statement is true.

f.

Expert Solution

To determine

To find: Whether the statement “If dimV=p and Span S=V , then S cannot be linearly dependent.” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that dimV=p . That is, the basis of the vector space contain p elements

Here, Span S=V , that is the set contain p elements span the p dimensional space.

Which implies the p elements is linearly independent.

Hence, the statement is false.

g.

Expert Solution

To determine

To find: Whether the statement “A plane in ℝ3 is a two-dimensional subspace” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Every plane in ℝ3 to be a two dimensional subspace is false.

Sometimes plane in ℝ3 does not form a vector space. Only the plane passing through origin form a subspace of ℝ3 .

Hence, the statement is false.

h.

Expert Solution

To determine

To find: Whether the statement “The non-pivot columns of a matrix are always linearly dependent” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Consider the matrix [1000001000000000] .

Here non-pivot columns are linearly independent.

Hence, the statement is false.

i.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix A can change the linear dependence relations among the rows of A” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Row operations on matrix A change the matrix. Thus, this changes the dependence relation among the rows of a matrix.

Hence, the statement is true.

j.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix can change the null space” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Row operations do not change the solution set of the system Ax=0 .

Therefore, row operations do not change the null space.

Hence, the statement is false.

k.

Expert Solution

To determine

To find: Whether the statement “The rank of a matrix equals the number of nonzero rows” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The rank of a matrix A is the dimension of the column space of A.

The dimension of column space of A is the number of pivot columns in A.

Therefore, the rank of matrix equals the number of pivot columns.

Consider the matrix [1111] .

The above matrix has 2 rows but rank of the matrix is 1.

Hence, the statement is false.

l.

Expert Solution

To determine

To find: Whether the statement “If an m×n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax=0 is m−k ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If U has k nonzero rows then, rank A=k .

According to the Rank Theorem, the rank of an m×n matrix A is same as the dimension of the column space of A and is equal to the number of pivot positions in A such that,

rank A+dimNul A=ndimNul A=n−rank A=n−k

Hence, the statement is false.

m.

Expert Solution

To determine

To find: Whether the statement “If B is obtained from a matrix A by several elementary row operations, then rank A=rank B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Elementary row operations does not change the number of pivot columns and hence, does not change the rank of a matrix.

Therefore, the rank of matrix B will be same as the rank of matrix A.

Hence, the statement is true.

n.

Expert Solution

To determine

To find: Whether the statement “The nonzero rows of a matrix A form a basis for Row A” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

To form a basis for A, the rows have to span A and should also be linearly independent.

The nonzero rows of a matrix A span Row A but that does not guarantee that they are linearly independent.

Hence, the statement is false.

o.

Expert Solution

To determine

To find: Whether the statement “If matrices A and B have the same reduced echelon form, then Row A=Row B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The nonzero rows of the echelon form of a matrix, form a basis for the row space of that matrix.

If the echelon form for two matrices is same, then the basis for the row spaces is also same.

Since, row spaces are vector spaces and if two vector spaces have same basis, then the vector spaces are same.

Hence, the statement is true.

p.

Expert Solution

To determine

To find: Whether the statement “If H is a subspace of ℝ3 , then there is a 3×3 matrix A such that H=Col A ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

If H is a zero, 1, 2, or 3 dimensional subspace of ℝ3 , there will always be a corresponding 3×3 matrix A formed by the basis of H.

The basis of H will then be in the column space of A.

Therefore, H=Col A .

Hence, the statement is true.

q.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and rank A=m , then the linear transformation x↦Ax is one-to-one” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

The transformation x↦Ax is one-to-one if and only if null space of A is zero.

Here, rank of the matrix is A thus, by the rank nullity theorem null space of A contain n−m elements.

Hence, the statement is false.

r.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and the linear transformation x↦Ax is onto, then rank A=m .)” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

If the transformation is onto then, Col A=ℝm

The rank of a matrix A is the dimension of the column space of A.

Therefore, the rank of A is m.

Hence, the statement is true.

s.

Expert Solution

To determine

To find: Whether the statement “A change-of-coordinate matrix is always invertible” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set.

The matrix PC←B is a square matrix thus; it must be an invertible matrix.

Hence, the statement is true.

t.

Expert Solution

To determine

To find: Whether the statement “If B={b1,…,bn} and C={c1,…,cn} are bases for a vector space V, then the j th column of the change-of-coordinates matrix PC←B is the coordinate vector [cj]B ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The jth column of the change-of-coordinates matrix PC←B is [bj]C .

Hence, the statement is false.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

a.

Expert Solution

To determine

To find: Whether the statement “The set of all linear combinations of v1,…,vp is a vector space” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here, the given vectors are v1,…,vp .

The span { v1,…,vp } always form a vector subspace of the vector space.

Thus, the linear combinations of v1,…,vp are also a vector space.

Hence, the statement is true.

b.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} spans V, then S spans V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The set S is S={v1,…,vp} .

It is given that the set {v1,…,vp−1} spans V. That is, span {v1,…,vp−1} = V .

That is, every element in the vector space V can be expressed by using the vectors {v1,…,vp−1} .

Here, the set S={v1,…,vp} , the set contain more than one element than the set {v1,…,vp−1} .

The smaller set {v1,…,vp−1} spans V which implies the bigger set S={v1,…,vp} also spans V.

If the vector space V is spanned by {v1,…,vp−1} , then its linear combinations also span V.

Hence, the statement is true.

c.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} is linearly independent, then so is S” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If {v1,…,vp−1} is a set of linearly independent vectors, then it does not imply that the set {v1,…,vp−1,vp} is also linearly independent as vp may be expressed as a linear combination of {v1,…,vp−1} .

Therefore, it does not imply that S is linearly independent.

Hence, the statement is false.

d.

Expert Solution

To determine

To find: Whether the statement “If S is linearly independent then S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

A set of vectors {v1,…,vp} is a basis for a vector space V if the following two conditions are satisfied:

1. The set of vectors {v1,…,vp} is a linearly independent set.

2. The set {v1,…,vp} spans the vector space V.

It can be seen that the second condition is not satisfied.

Hence, the statement is false.

e.

Expert Solution

To determine

To find: Whether the statement “If Span S=V , then some subset of S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that the vector space V is spanned by S, which is nonzero set.

Suppose the set S is linearly independent then, the set S form a basis for V.

Suppose the set S is linearly dependent then, some subset of S linearly independent and which spans V.

That is, some subset of S form a basis for V.

Hence, the statement is true.

f.

Expert Solution

To determine

To find: Whether the statement “If dimV=p and Span S=V , then S cannot be linearly dependent.” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that dimV=p . That is, the basis of the vector space contain p elements

Here, Span S=V , that is the set contain p elements span the p dimensional space.

Which implies the p elements is linearly independent.

Hence, the statement is false.

g.

Expert Solution

To determine

To find: Whether the statement “A plane in ℝ3 is a two-dimensional subspace” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Every plane in ℝ3 to be a two dimensional subspace is false.

Sometimes plane in ℝ3 does not form a vector space. Only the plane passing through origin form a subspace of ℝ3 .

Hence, the statement is false.

h.

Expert Solution

To determine

To find: Whether the statement “The non-pivot columns of a matrix are always linearly dependent” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Consider the matrix [1000001000000000] .

Here non-pivot columns are linearly independent.

Hence, the statement is false.

i.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix A can change the linear dependence relations among the rows of A” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Row operations on matrix A change the matrix. Thus, this changes the dependence relation among the rows of a matrix.

Hence, the statement is true.

j.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix can change the null space” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Row operations do not change the solution set of the system Ax=0 .

Therefore, row operations do not change the null space.

Hence, the statement is false.

k.

Expert Solution

To determine

To find: Whether the statement “The rank of a matrix equals the number of nonzero rows” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The rank of a matrix A is the dimension of the column space of A.

The dimension of column space of A is the number of pivot columns in A.

Therefore, the rank of matrix equals the number of pivot columns.

Consider the matrix [1111] .

The above matrix has 2 rows but rank of the matrix is 1.

Hence, the statement is false.

l.

Expert Solution

To determine

To find: Whether the statement “If an m×n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax=0 is m−k ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If U has k nonzero rows then, rank A=k .

According to the Rank Theorem, the rank of an m×n matrix A is same as the dimension of the column space of A and is equal to the number of pivot positions in A such that,

rank A+dimNul A=ndimNul A=n−rank A=n−k

Hence, the statement is false.

m.

Expert Solution

To determine

To find: Whether the statement “If B is obtained from a matrix A by several elementary row operations, then rank A=rank B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Elementary row operations does not change the number of pivot columns and hence, does not change the rank of a matrix.

Therefore, the rank of matrix B will be same as the rank of matrix A.

Hence, the statement is true.

n.

Expert Solution

To determine

To find: Whether the statement “The nonzero rows of a matrix A form a basis for Row A” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

To form a basis for A, the rows have to span A and should also be linearly independent.

The nonzero rows of a matrix A span Row A but that does not guarantee that they are linearly independent.

Hence, the statement is false.

o.

Expert Solution

To determine

To find: Whether the statement “If matrices A and B have the same reduced echelon form, then Row A=Row B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The nonzero rows of the echelon form of a matrix, form a basis for the row space of that matrix.

If the echelon form for two matrices is same, then the basis for the row spaces is also same.

Since, row spaces are vector spaces and if two vector spaces have same basis, then the vector spaces are same.

Hence, the statement is true.

p.

Expert Solution

To determine

To find: Whether the statement “If H is a subspace of ℝ3 , then there is a 3×3 matrix A such that H=Col A ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

If H is a zero, 1, 2, or 3 dimensional subspace of ℝ3 , there will always be a corresponding 3×3 matrix A formed by the basis of H.

The basis of H will then be in the column space of A.

Therefore, H=Col A .

Hence, the statement is true.

q.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and rank A=m , then the linear transformation x↦Ax is one-to-one” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

The transformation x↦Ax is one-to-one if and only if null space of A is zero.

Here, rank of the matrix is A thus, by the rank nullity theorem null space of A contain n−m elements.

Hence, the statement is false.

r.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and the linear transformation x↦Ax is onto, then rank A=m .)” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

If the transformation is onto then, Col A=ℝm

The rank of a matrix A is the dimension of the column space of A.

Therefore, the rank of A is m.

Hence, the statement is true.

s.

Expert Solution

To determine

To find: Whether the statement “A change-of-coordinate matrix is always invertible” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set.

The matrix PC←B is a square matrix thus; it must be an invertible matrix.

Hence, the statement is true.

t.

Expert Solution

To determine

To find: Whether the statement “If B={b1,…,bn} and C={c1,…,cn} are bases for a vector space V, then the j th column of the change-of-coordinates matrix PC←B is the coordinate vector [cj]B ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The jth column of the change-of-coordinates matrix PC←B is [bj]C .

Hence, the statement is false.

Answer 8

a.

Expert Solution

To determine

To find: Whether the statement “The set of all linear combinations of v1,…,vp is a vector space” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here, the given vectors are v1,…,vp .

The span { v1,…,vp } always form a vector subspace of the vector space.

Thus, the linear combinations of v1,…,vp are also a vector space.

Hence, the statement is true.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

To find: Whether the statement “The set of all linear combinations of v1,…,vp is a vector space” is true or false.

Answer 13

Answer to Problem 1SE

The statement is true.

Answer 14

Explanation of Solution

Here, the given vectors are v1,…,vp .

The span { v1,…,vp } always form a vector subspace of the vector space.

Thus, the linear combinations of v1,…,vp are also a vector space.

Hence, the statement is true.

Answer 15

b.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} spans V, then S spans V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The set S is S={v1,…,vp} .

It is given that the set {v1,…,vp−1} spans V. That is, span {v1,…,vp−1} = V .

That is, every element in the vector space V can be expressed by using the vectors {v1,…,vp−1} .

Here, the set S={v1,…,vp} , the set contain more than one element than the set {v1,…,vp−1} .

The smaller set {v1,…,vp−1} spans V which implies the bigger set S={v1,…,vp} also spans V.

If the vector space V is spanned by {v1,…,vp−1} , then its linear combinations also span V.

Hence, the statement is true.

Answer 16

b.

Expert Solution

Answer 17

b.

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

To find: Whether the statement “If {v1,…,vp−1} spans V, then S spans V” is true or false.

Answer 20

Answer to Problem 1SE

The statement is true.

Answer 21

Explanation of Solution

The set S is S={v1,…,vp} .

It is given that the set {v1,…,vp−1} spans V. That is, span {v1,…,vp−1} = V .

That is, every element in the vector space V can be expressed by using the vectors {v1,…,vp−1} .

Here, the set S={v1,…,vp} , the set contain more than one element than the set {v1,…,vp−1} .

The smaller set {v1,…,vp−1} spans V which implies the bigger set S={v1,…,vp} also spans V.

If the vector space V is spanned by {v1,…,vp−1} , then its linear combinations also span V.

Hence, the statement is true.

Answer 22

c.

Expert Solution

To determine

To find: Whether the statement “If {v1,…,vp−1} is linearly independent, then so is S” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If {v1,…,vp−1} is a set of linearly independent vectors, then it does not imply that the set {v1,…,vp−1,vp} is also linearly independent as vp may be expressed as a linear combination of {v1,…,vp−1} .

Therefore, it does not imply that S is linearly independent.

Hence, the statement is false.

Answer 23

c.

Expert Solution

Answer 24

c.

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

To find: Whether the statement “If {v1,…,vp−1} is linearly independent, then so is S” is true or false.

Answer 27

Answer to Problem 1SE

The statement is false.

Answer 28

Explanation of Solution

If {v1,…,vp−1} is a set of linearly independent vectors, then it does not imply that the set {v1,…,vp−1,vp} is also linearly independent as vp may be expressed as a linear combination of {v1,…,vp−1} .

Therefore, it does not imply that S is linearly independent.

Hence, the statement is false.

Answer 29

d.

Expert Solution

To determine

To find: Whether the statement “If S is linearly independent then S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

A set of vectors {v1,…,vp} is a basis for a vector space V if the following two conditions are satisfied:

1. The set of vectors {v1,…,vp} is a linearly independent set.

2. The set {v1,…,vp} spans the vector space V.

It can be seen that the second condition is not satisfied.

Hence, the statement is false.

Answer 30

d.

Expert Solution

Answer 31

d.

Expert Solution

Answer 32

Expert Solution

Answer 33

To determine

To find: Whether the statement “If S is linearly independent then S is a basis for V” is true or false.

Answer 34

Answer to Problem 1SE

The statement is false.

Answer 35

Explanation of Solution

A set of vectors {v1,…,vp} is a basis for a vector space V if the following two conditions are satisfied:

1. The set of vectors {v1,…,vp} is a linearly independent set.

2. The set {v1,…,vp} spans the vector space V.

It can be seen that the second condition is not satisfied.

Hence, the statement is false.

Answer 36

e.

Expert Solution

To determine

To find: Whether the statement “If Span S=V , then some subset of S is a basis for V” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that the vector space V is spanned by S, which is nonzero set.

Suppose the set S is linearly independent then, the set S form a basis for V.

Suppose the set S is linearly dependent then, some subset of S linearly independent and which spans V.

That is, some subset of S form a basis for V.

Hence, the statement is true.

Answer 37

e.

Expert Solution

Answer 38

e.

Expert Solution

Answer 39

Expert Solution

Answer 40

To determine

To find: Whether the statement “If Span S=V , then some subset of S is a basis for V” is true or false.

Answer 41

Answer to Problem 1SE

The statement is true.

Answer 42

Explanation of Solution

It is given that the vector space V is spanned by S, which is nonzero set.

Suppose the set S is linearly independent then, the set S form a basis for V.

Suppose the set S is linearly dependent then, some subset of S linearly independent and which spans V.

That is, some subset of S form a basis for V.

Hence, the statement is true.

Answer 43

f.

Expert Solution

To determine

To find: Whether the statement “If dimV=p and Span S=V , then S cannot be linearly dependent.” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

It is given that dimV=p . That is, the basis of the vector space contain p elements

Here, Span S=V , that is the set contain p elements span the p dimensional space.

Which implies the p elements is linearly independent.

Hence, the statement is false.

Answer 44

f.

Expert Solution

Answer 45

f.

Expert Solution

Answer 46

Expert Solution

Answer 47

To determine

To find: Whether the statement “If dimV=p and Span S=V , then S cannot be linearly dependent.” is true or false.

Answer 48

Answer to Problem 1SE

The statement is true.

Answer 49

Explanation of Solution

It is given that dimV=p . That is, the basis of the vector space contain p elements

Here, Span S=V , that is the set contain p elements span the p dimensional space.

Which implies the p elements is linearly independent.

Hence, the statement is false.

Answer 50

g.

Expert Solution

To determine

To find: Whether the statement “A plane in ℝ3 is a two-dimensional subspace” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Every plane in ℝ3 to be a two dimensional subspace is false.

Sometimes plane in ℝ3 does not form a vector space. Only the plane passing through origin form a subspace of ℝ3 .

Hence, the statement is false.

Answer 51

g.

Expert Solution

Answer 52

g.

Expert Solution

Answer 53

Expert Solution

Answer 54

To determine

To find: Whether the statement “A plane in ℝ3 is a two-dimensional subspace” is true or false.

Answer 55

Answer to Problem 1SE

The statement is false.

Answer 56

Explanation of Solution

Every plane in ℝ3 to be a two dimensional subspace is false.

Sometimes plane in ℝ3 does not form a vector space. Only the plane passing through origin form a subspace of ℝ3 .

Hence, the statement is false.

Answer 57

h.

Expert Solution

To determine

To find: Whether the statement “The non-pivot columns of a matrix are always linearly dependent” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Consider the matrix [1000001000000000] .

Here non-pivot columns are linearly independent.

Hence, the statement is false.

Answer 58

h.

Expert Solution

Answer 59

h.

Expert Solution

Answer 60

Expert Solution

Answer 61

To determine

To find: Whether the statement “The non-pivot columns of a matrix are always linearly dependent” is true or false.

Answer 62

Answer to Problem 1SE

The statement is false.

Answer 63

Explanation of Solution

Consider the matrix [1000001000000000] .

Here non-pivot columns are linearly independent.

Hence, the statement is false.

Answer 64

i.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix A can change the linear dependence relations among the rows of A” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Row operations on matrix A change the matrix. Thus, this changes the dependence relation among the rows of a matrix.

Hence, the statement is true.

Answer 65

i.

Expert Solution

Answer 66

i.

Expert Solution

Answer 67

Expert Solution

Answer 68

To determine

To find: Whether the statement “Row operations on a matrix A can change the linear dependence relations among the rows of A” is true or false.

Answer 69

Answer to Problem 1SE

The statement is true.

Answer 70

Explanation of Solution

Row operations on matrix A change the matrix. Thus, this changes the dependence relation among the rows of a matrix.

Hence, the statement is true.

Answer 71

j.

Expert Solution

To determine

To find: Whether the statement “Row operations on a matrix can change the null space” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Row operations do not change the solution set of the system Ax=0 .

Therefore, row operations do not change the null space.

Hence, the statement is false.

Answer 72

j.

Expert Solution

Answer 73

j.

Expert Solution

Answer 74

Expert Solution

Answer 75

To determine

To find: Whether the statement “Row operations on a matrix can change the null space” is true or false.

Answer 76

Answer to Problem 1SE

The statement is false.

Answer 77

Explanation of Solution

Row operations do not change the solution set of the system Ax=0 .

Therefore, row operations do not change the null space.

Hence, the statement is false.

Answer 78

k.

Expert Solution

To determine

To find: Whether the statement “The rank of a matrix equals the number of nonzero rows” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The rank of a matrix A is the dimension of the column space of A.

The dimension of column space of A is the number of pivot columns in A.

Therefore, the rank of matrix equals the number of pivot columns.

Consider the matrix [1111] .

The above matrix has 2 rows but rank of the matrix is 1.

Hence, the statement is false.

Answer 79

k.

Expert Solution

Answer 80

k.

Expert Solution

Answer 81

Expert Solution

Answer 82

To determine

To find: Whether the statement “The rank of a matrix equals the number of nonzero rows” is true or false.

Answer 83

Answer to Problem 1SE

The statement is false.

Answer 84

Explanation of Solution

The rank of a matrix A is the dimension of the column space of A.

The dimension of column space of A is the number of pivot columns in A.

Therefore, the rank of matrix equals the number of pivot columns.

Consider the matrix [1111] .

The above matrix has 2 rows but rank of the matrix is 1.

Hence, the statement is false.

Answer 85

l.

Expert Solution

To determine

To find: Whether the statement “If an m×n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax=0 is m−k ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

If U has k nonzero rows then, rank A=k .

According to the Rank Theorem, the rank of an m×n matrix A is same as the dimension of the column space of A and is equal to the number of pivot positions in A such that,

rank A+dimNul A=ndimNul A=n−rank A=n−k

Hence, the statement is false.

Answer 86

l.

Expert Solution

Answer 87

l.

Expert Solution

Answer 88

Expert Solution

Answer 89

To determine

To find: Whether the statement “If an m×n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax=0 is m−k ” is true or false.

Answer 90

Answer to Problem 1SE

The statement is false.

Answer 91

Explanation of Solution

If U has k nonzero rows then, rank A=k .

According to the Rank Theorem, the rank of an m×n matrix A is same as the dimension of the column space of A and is equal to the number of pivot positions in A such that,

rank A+dimNul A=ndimNul A=n−rank A=n−k

Hence, the statement is false.

Answer 92

m.

Expert Solution

To determine

To find: Whether the statement “If B is obtained from a matrix A by several elementary row operations, then rank A=rank B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Elementary row operations does not change the number of pivot columns and hence, does not change the rank of a matrix.

Therefore, the rank of matrix B will be same as the rank of matrix A.

Hence, the statement is true.

Answer 93

m.

Expert Solution

Answer 94

m.

Expert Solution

Answer 95

Expert Solution

Answer 96

To determine

To find: Whether the statement “If B is obtained from a matrix A by several elementary row operations, then rank A=rank B ” is true or false.

Answer 97

Answer to Problem 1SE

The statement is true.

Answer 98

Explanation of Solution

Elementary row operations does not change the number of pivot columns and hence, does not change the rank of a matrix.

Therefore, the rank of matrix B will be same as the rank of matrix A.

Hence, the statement is true.

Answer 99

n.

Expert Solution

To determine

To find: Whether the statement “The nonzero rows of a matrix A form a basis for Row A” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

To form a basis for A, the rows have to span A and should also be linearly independent.

The nonzero rows of a matrix A span Row A but that does not guarantee that they are linearly independent.

Hence, the statement is false.

Answer 100

n.

Expert Solution

Answer 101

n.

Expert Solution

Answer 102

Expert Solution

Answer 103

To determine

To find: Whether the statement “The nonzero rows of a matrix A form a basis for Row A” is true or false.

Answer 104

Answer to Problem 1SE

The statement is false.

Answer 105

Explanation of Solution

To form a basis for A, the rows have to span A and should also be linearly independent.

The nonzero rows of a matrix A span Row A but that does not guarantee that they are linearly independent.

Hence, the statement is false.

Answer 106

o.

Expert Solution

To determine

To find: Whether the statement “If matrices A and B have the same reduced echelon form, then Row A=Row B ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The nonzero rows of the echelon form of a matrix, form a basis for the row space of that matrix.

If the echelon form for two matrices is same, then the basis for the row spaces is also same.

Since, row spaces are vector spaces and if two vector spaces have same basis, then the vector spaces are same.

Hence, the statement is true.

Answer 107

o.

Expert Solution

Answer 108

o.

Expert Solution

Answer 109

Expert Solution

Answer 110

To determine

To find: Whether the statement “If matrices A and B have the same reduced echelon form, then Row A=Row B ” is true or false.

Answer 111

Answer to Problem 1SE

The statement is true.

Answer 112

Explanation of Solution

The nonzero rows of the echelon form of a matrix, form a basis for the row space of that matrix.

If the echelon form for two matrices is same, then the basis for the row spaces is also same.

Since, row spaces are vector spaces and if two vector spaces have same basis, then the vector spaces are same.

Hence, the statement is true.

Answer 113

p.

Expert Solution

To determine

To find: Whether the statement “If H is a subspace of ℝ3 , then there is a 3×3 matrix A such that H=Col A ” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

If H is a zero, 1, 2, or 3 dimensional subspace of ℝ3 , there will always be a corresponding 3×3 matrix A formed by the basis of H.

The basis of H will then be in the column space of A.

Therefore, H=Col A .

Hence, the statement is true.

Answer 114

p.

Expert Solution

Answer 115

p.

Expert Solution

Answer 116

Expert Solution

Answer 117

To determine

To find: Whether the statement “If H is a subspace of ℝ3 , then there is a 3×3 matrix A such that H=Col A ” is true or false.

Answer 118

Answer to Problem 1SE

The statement is true.

Answer 119

Explanation of Solution

If H is a zero, 1, 2, or 3 dimensional subspace of ℝ3 , there will always be a corresponding 3×3 matrix A formed by the basis of H.

The basis of H will then be in the column space of A.

Therefore, H=Col A .

Hence, the statement is true.

Answer 120

q.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and rank A=m , then the linear transformation x↦Ax is one-to-one” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

The transformation x↦Ax is one-to-one if and only if null space of A is zero.

Here, rank of the matrix is A thus, by the rank nullity theorem null space of A contain n−m elements.

Hence, the statement is false.

Answer 121

q.

Expert Solution

Answer 122

q.

Expert Solution

Answer 123

Expert Solution

Answer 124

To determine

To find: Whether the statement “If A is m×n and rank A=m , then the linear transformation x↦Ax is one-to-one” is true or false.

Answer 125

Answer to Problem 1SE

The statement is false.

Answer 126

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

The transformation x↦Ax is one-to-one if and only if null space of A is zero.

Here, rank of the matrix is A thus, by the rank nullity theorem null space of A contain n−m elements.

Hence, the statement is false.

Answer 127

r.

Expert Solution

To determine

To find: Whether the statement “If A is m×n and the linear transformation x↦Ax is onto, then rank A=m .)” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

If the transformation is onto then, Col A=ℝm

The rank of a matrix A is the dimension of the column space of A.

Therefore, the rank of A is m.

Hence, the statement is true.

Answer 128

r.

Expert Solution

Answer 129

r.

Expert Solution

Answer 130

Expert Solution

Answer 131

To determine

To find: Whether the statement “If A is m×n and the linear transformation x↦Ax is onto, then rank A=m .)” is true or false.

Answer 132

Answer to Problem 1SE

The statement is true.

Answer 133

Explanation of Solution

Here the matrix A is linear transformation from Rm to Rn .

If the transformation is onto then, Col A=ℝm

The rank of a matrix A is the dimension of the column space of A.

Therefore, the rank of A is m.

Hence, the statement is true.

Answer 134

s.

Expert Solution

To determine

To find: Whether the statement “A change-of-coordinate matrix is always invertible” is true or false.

Answer to Problem 1SE

The statement is true.

Explanation of Solution

The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set.

The matrix PC←B is a square matrix thus; it must be an invertible matrix.

Hence, the statement is true.

Answer 135

s.

Expert Solution

Answer 136

s.

Expert Solution

Answer 137

Expert Solution

Answer 138

To determine

To find: Whether the statement “A change-of-coordinate matrix is always invertible” is true or false.

Answer 139

Answer to Problem 1SE

The statement is true.

Answer 140

Explanation of Solution

The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set.

The matrix PC←B is a square matrix thus; it must be an invertible matrix.

Hence, the statement is true.

Answer 141

t.

Expert Solution

To determine

To find: Whether the statement “If B={b1,…,bn} and C={c1,…,cn} are bases for a vector space V, then the j th column of the change-of-coordinates matrix PC←B is the coordinate vector [cj]B ” is true or false.

Answer to Problem 1SE

The statement is false.

Explanation of Solution

The jth column of the change-of-coordinates matrix PC←B is [bj]C .

Hence, the statement is false.

Answer 142

t.

Expert Solution

Answer 143

t.

Expert Solution

Answer 144

Expert Solution

Answer 145

To determine

To find: Whether the statement “If B={b1,…,bn} and C={c1,…,cn} are bases for a vector space V, then the j th column of the change-of-coordinates matrix PC←B is the coordinate vector [cj]B ” is true or false.

Answer 146

Answer to Problem 1SE

The statement is false.

Answer 147

Explanation of Solution

The jth column of the change-of-coordinates matrix PC←B is [bj]C .

Hence, the statement is false.

Answer 148

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