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1. [Normal equations, QR decomposition, Householder reflections, (a) Let A E Rmx" with m > n. Show that AA is positive definite if and only if A has full column rank.1 (b) Let QR = [A b] c Remx(":+1) be the reduced QR decomposition of the matrix that has a columns the columns of A and additional the column b. Assume the diagonal of R is positive. The matrix R has the the elements R = R r with RE RX, r ER", p E R. Show that for the least-squares problem min | |Ax - bl|2, it holds Rx = r and p = ||Ax - bl|2