quiz1 - Quiz #1 1. Label the following statements as being...

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Unformatted text preview: Quiz #1 1. Label the following statements as being true or false. No verification is needed. (a) A vector space may have more than one zero vector. (Solution) False. (b) If S is a subset of a vector space V , then span(S) equals the intersection of all subspaces of V that contain S. (Solution) True. (c) Subsets of linearly dependent sets are linearly dependent. (Solution) False. (d) The dimension of Mm×n (F ) is m + n. (Solution) False. (e) Let T : V → W be a linear transformation of vector spaces. Then T is one-to-one if and only if N (T ) = {0}. (Solution) True. 2. Determine whether the following set is a basis for R3 or not. {(1, −3, −2), (−3, 1, 3), (−2, −10, −2)}. (Solution) The above set is not a basis for R3 since they are linearly dependent; 1 −3 −2 det −3 1 −10 = 0. −2 3 −2 3. A matrix A is called skew-symmetric if At = −A. The set of all skewsymmetric 3 × 3 matrices is a subspace W of M3×3 (R) (You need NOT prove it). What is the dimension of W ? (Solution) Let a b c A = d e f . g h i 1 be a skew-symmetric matrix. Then, since At = −A, d = −b, g = −c, h = −f. a = e = i = 0, Therefore, 0 1 0 0 0 1 0 0 0 A = b −1 0 0 + c 0 0 0 + f 0 0 1 . 0 0 0 −1 0 0 0 −1 0 Thus, the subset 0 0 1 0 0 0 0 1 0 −1 0 0 , 0 0 0 , 0 0 1 0 0 0 −1 0 0 0 −1 0 is a basis for W , and hence dim W = 3. [A generalized version] A matrix A is called skew-symmetric if At = −A. The set of all skew-symmetric n × n matrices is a subspace W of Mn×n (R). What is the dimension of W ? (Solution) Let A be skew-symmetric. Then (At )ij = (−A)ij for all i, j = 1, . . . , n. So Aji = −Aij . Therefore Aii = 0 for all i = 0, . . . , n and n A= Aij (Eij − Eji ). Aij Eij = i,j=1 1≤j<i≤n Thus, the set {Eij − Eji | 1 ≤ j < i ≤ n} is a basis for W and n 2 dim W = 2 = n(n − 1) . 2 ...
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quiz1 - Quiz #1 1. Label the following statements as being...

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