MATH 401 Lecture Notes - Lecture 5: Gaussian Elimination
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Solutions to ma242 quiz 8, 11/07/06: let h = spanfv1; v2; v3; v4g, where v1 = 0. A (a) explain why h is a subspace of r3. (justify your answer!) (b) find a basis for h. (c) using your answer in (b), explain why h is isomorphic to r2. (justify your answer!) A = [v1 : : : v4] = 2. Hence, h is isomorphic to r2 under the coordinate mapping, see theorem 8 in section. Solution: (a) the change-of-coordinates matrix pb = [b1 b2 b3] is row-equivalent to the identity matrix i3. Hence, by the invertible matrix theorem, pb is invertible, and its columns form a basis for r3. (b) by (a), it follows that. 5 for the change-of-coordinates matrix pb, with [x]e = x = pb[x]b. (c) to solve the equation [x]b = p (cid:0)1. B x for [x]b, row reduce the augmented matrix [pb x]:
