Problem 7. Express the rational function (x - 1)(r + 3) as a linear combination of the rational functions I + 3 Problem 8. Express the polynomial 1 + 2r + 3x- + 23 as a linear combination of the polynomials 1- x, 1+x, I+atr', 1+x+x3+x Problem 9. Suppose T : R" - Rm and S : Rm _+ R* are linear mappings. Show that So T is linear. Problem 10. Show that the mapping A : R2 x R2 - R defined by ((a, b), (c, d)) + ad - be is bilinear. Show that A(v, w) = -A(w, v) for all v, w in R2. Show that the corresponding mapping C x C - C is given by (z, w) Im(Ew) and show that this is invariant under rotation about the origin. That is, if v, w are in R? and v' and w are their images under rotation by some amount 0, then show that A(v, w) = A(v', w'). Why is this extra-easy with complex numbers? Problem 11. a) Let u = (1, 2, 1) and v = (2, 2, 4). Find the linear combination ou + By that's as close as possible to w = (3, 2, 1) in RS. b) Find the least squares best-fit line for the data set { (1, 2), (2,5), (3, 4), (4, 6)} (plot everything, of course). Explain your set-up and the connection with the previous part of this