Math 136 Spring 2017: Assignment 2 (Due Wednesday, May 17, 8pm) Topics: Section 1.1, 1.3, 1.4, bases, subspaces, dot product, norms. 1. Determine with proof which of the following are subspaces of R3 and which are not. (a) S = 1 R3 : = 0 (b) S = R3 : + = 0 0 (c) S = 0 0 (d) S = R3 : = 0 0 1 x1 2 , 2 is a basis of S. x2 : 2x1 + x2 2x3 = 0 . Show that B = 2. Let S = 0 1 x3 x1 3. Find a basis for the subspace S = x2 : x1 + 2x2 = x3 of R3 . x3 1 2 4. Let u ~ = 3 and ~v = 1. Evaluate the following. 1 1 (a) u ~ ~v . (b) k~uk 5. Suppose u ~ , ~v Rn are such that k~u + ~v k2 = k~uk2 + k~vk2 . Prove that u ~ ~v = 0. 1