CS 660 Fall 2023 - HW#1 Linear Algebra Due Sep 21, 2023 Complete all five problems. Put your pages in order and scan your solutions and upload one PDF. I will not grade multiple files, jpegs, Mac .Pages, etc. Each problem is worth 4 points for a total of 20 points. 1. Compute the following matrix products, if possible: (a) (b) 456 0 0 (c) baw - O O - VA (d) - N A N A - UN - O N - W -1 (e) A - 2 UN - N - J 1 - 4 [4 points] 2. Write y = as a linear combination of x1 = *2 = 2 . 83 = -1 3. Consider two subspaces Uj and U2, where U1 is the solution space of the homogeneous equation system A j* = 0 and U2 is the solution space of the homogeneous equation system A2x = 0 with IN A1 = NWO A2 = W . (a) Determine the dimension of U1, U2 (b) Determine bases of U1 and U2 (c) Determine a basis of U1 n U2 4. Suppose S = {v1, V2, . .., Vm) spans a vector space V. Prove: (a) If we V, then {w, v1, V2, . .., Vm) is linearly dependent and spans V. (b) If v; is a linear combination of (v1, v2, . .., Vi-1), then S without v; spans V. [4 points] 5. Consider the basis of R'. Find the change-of-basis matrix P from the standard basis (e1, e2, e3) to the basis B. [4 points]