UWthat L(Jf(x))= f(x) f(0) is a linear operator in C[-1,1. (b) Find ker L above. Find the range of L above. 10. Let S = {(x1,x2, x3, 4) 1+2 x3 +4} be a subspace of R4. Find S-. 11. Given v =(1,-1, 1,...

#12 a,b

Extracted text: UWthat L(Jf(x))= f(x) f(0) is a linear operator in C[-1,1. (b) Find ker L above. Find the range of L above. 10. Let S = {(x1,x2, x3, 4) 1+2 x3 +4} be a subspace of R4. Find S-. 11. Given v =(1,-1, 1, 1) and w = (4, 2,2,1) (a) Determine the angle between v and w. b Find the orthogonal complement of V span fv, w}. 12. Let A be an m x n matrix. (a) Suppose that rank A = r, what are dimensions of N(A) and N(A). Verify that N(AT A) = N(A) and rank(AT A) =r.