For matrix A assume that (ATA)-¹ exists. The projection matrix onto the column space of A is P = A(ATA)-¹AT. (We haven't defined column space yet but that is not needed here.) (a) Prove that P² = P. = = = (b) Prove that for all 6, Añ 6 has a solution & if and only if P₁ = 6 has a solution 7. That is show that if there exists a ū such that Pu 6 then there exists a v such that Av 6 and conversely show that if there exists a v such that Av = 6 then there exists a ū such that Pū = b. Use the given information to describe a v in terms of u and A that 'works' and for the converse describe a ū in terms of 7 and A that 'works'.

For both of the following make use of the matrix multiplication definition: If A is m × n and B is n xp then (AB)ij = Σ=1(A)ik (B)kj. (a) The trace of a square matrix is the sum of its diagonal elements. Writing Tr(A) for the trace of an n × n matrix this is Tr(A) = 1 A. Prove that Tr(AB) = Tr(BA). Assume that A is m × n, then B is n × m in order for both to be defined. (b) If L and M are n×n lower triangular matrices, prove that the product LM is lower triangular. A matrix A is lower triangular if (A)ij O whenever j > i. =

17?module_item_id=4856618 was.pdx. Given = 0.1714 and n = 35 for the high income group, Test the claim that the proportion of children in the high income group that drew a picture of the nickel too large is smaller than 50%. Test at the 0.03 significance level. a. Identify the correct alternative hypothesis: Op = .50 Op.50 Oμ=.50 Op>.50 Op <.50 b. The test statistic formula set up with numbers is as follows: Provide all the decimal places. p-P x= = p-(1-p) 16 (1- ))/) The final answer for the test statistic from technology (Excel or Ti Calculator functions NOT from the above formula) is as follows: Round to 4 decimal places. x = c. Using the P-value method, the P-value is: Round to 4 decimal places. d. Based on this, we O Reject Ho O Fail to reject Ho e. Which means O There is not sufficient evidence to support the claim O There is not sufficient evidence to warrant rejection of the claim O There is sufficient evidence to warrant rejection of the claim O The sample data supports the…