1. [5] Prove that the matrix M = 3 CO is not positive definite. (This is equivalent to asking for a column vector v such that v * 0 and UT Mv = 0.)2. [5] Determine the matrix that represents orthogonal projection from R3 onto the line L E R spanned by 'U = (1,2,3, 4,5,6, 7, 8) with respect to the standard basis of R3. (Suggestion: This matrix has a lot of fractions, but you can factor the common denominator out, leaving a matrix with integer entries. You don't have to Show your work if you can clearly explain your work.) 3. [15] Work one of the following. (a) Assuming the matrix MM is invertible, prove that w = (MTM)-1(MTb) is the vector closest to the solution of the linear system Mv = b. (This is equivalent to showing that || Mw - bl| R by g( A, B) = trace(AT B). Prove that g is an inner product on Roxq. (c) Apply the Gram-Schmidt algorithm to {1, 2t, 6t2, 20t3}, where the inner product is given by (f,g) = / f(t)g(t) dt. (Suggestions: Denote fo = 1, f1 = t, f2 = t2, f3 = to, and let go, 91, 92, and 93 be the results of the iterations of the algorithm. Thus, in the Oth iteration, we have go = fo = 1. You can check your work after each iteration by verifying (gi, g;) = 0 whenever i * j. Notice that fot* dt = 1, for all k = 0, 1, 2, 3, ....)