MATH115 Study Guide - Quiz Guide: Hyperplane, Dot Product, Linear Combination
2 Dec 2014
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Math 115 fall 2012 lab 2 solutions: {6 marks} determine whether or not the following sets are subspaces in their respective vector spaces. If so, prove it using the de nition of subspaces. If not, provide a counterexample where a rule of subspaces is violated. (a) {(cid:126)x r3 | x1 2x2 = 0, 3x1 x3 = 0}. This is non-empty since (cid:126)0 is in the set. Suppose (cid:126)u and (cid:126)v are vectors in the set. Then, by de nition, u1 2u2 = 0, and 3v1 v3 = 0. Now let (cid:126)z = (cid:126)u + (cid:126)v. then z1 = u1 + v1, z2 = u2 + v2 and z3 = u3 + v3. Observe that: z1 2z2 = u1 + v1 2(u2 + v2) = u1 2u2 + v1 2v2 = 0 + 0 = 0. 3z1 z3 = 3(u1 + v1) u3 v3 = 3u1 u3 + 3v1 v3 = 0 + 0 = 0.
