For Problem #12, how do I prove that the set is a basis for V? I think that infinity is the basis, but I'm not sure. This is a Linear Algebra type of question. Here is a picture.
Extracted text: ing techniques from Chapter 5, we can show that det(C) = -59. Since the minant of the coefficient matrix C is nonzero, it follows from Theorem 5.7 (the Big rem, Version 7) in Section 5.2 that our linear system has only the trivial solution. efore V is a linearly independent set and is a basis for P. DAVID I the set V could pos- 10, VRx2 11. V- ((1,0, 0, 0, 0, ...), (1,-1, 0, 0, 0, ...) (1,-1, 1, 0,0,.), (1,-1, 1,-1,0,...),. ., VR 12. (1, x+ 1, x+x+1, x+ x2+ x+ 1. ). V=P V P VR2 For Exercises 13-18, find a basis for the subspace S and determine dim(S). 13. S is the subspace of R consisting of matrices with trace equal to zero, (The trace is the sum of the diagonal terms of a matrix.) VRx2 14. S is the subspace of P2 consisting of polynomials with graphs crossing the origin. 15. S is the subspace of R2x2 consisting of matrices with compo- basis for V. nents that add to zero. 16. S is the subspace of T(2, 2) consisting of linear transforma- tions T: R R2 such that T (x) = ax for some scalar a. + 1, V= P 17. S is the subspace of T(2, 2) consisting of linear transforma- tions T: R VRx2 R such that T(v) = 0 for a specific vector v.