Problem 2. In this problem, whenever you need to solve a system of linear equations, be sure to use augmented matrices and Gaussian elimination! Show your work (and use an online solver to check it). (a) Let X, Y, Z E R". Use the definition of linear independence to show that if {X, Y, Z } is a linearly independent set, then {2X + 3Y, X + 2Y + Z, -X +Y + Z} is also a linearly independent set. (b) Let X, Y E R". Show that | |X + Y||? = | |X||2 + | |Y||2 if and only if X and Y are orthogonal. (c) Let X = [1, 2,0, -1]] and Y = [0, -2, 1,3]]. Find a vector Z parallel to vector Y such that the distance d( X, Z) is as small as possible. (d) Let X1 = [1, 0, -1, 3]], X2 = [1, -3, 1,0]7, and X3 = [0, -1, -3, -1] be vectors in R4. (i) Show that {X1, X2, X3} is an orthogonal set. (ii) Find an orthogonal basis R' that contains X1, X2, X3. (iii) Write the vector X = [3, -2,2, 7]" as a linear combination of the vectors in your orthogonal basis from (ii) . (e) Let U = span {X1, X2, X3), where X1 = [1, -1, -1]7, X2 = [2, 1, 4]], and X3 = [5, -4, -3] are vectors in R3. Using the Gram-Schmidt Algorithm, find an orthonormal basis for the subspace U