Adv Alegbra HW Solutions 132

Adv Alegbra HW Solutions 132 - 132 (ii) Every matrix over R...

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Unformatted text preview: 132 (ii) Every matrix over R is similar to infinitely many different matrices. Solution. False. (iii) If S and T are linear transformations on the plane R2 that agree on two nonzero points, then S = T . Solution. True. (iv) If A and B are n × n nonsingular matrices, then A + B is nonsingular. Solution. False. (v) If A and B are n × n nonsingular matrices, then AB is nonsingular. Solution. True. (vi) If k is a field, then { A ∈ Matn (k ) : AB = B A for all B ∈ Matn (k )} is a 1-dimensional subspace of Matn (k ). Solution. True. (vii) The vector space of all 3 × 3 symmetric matrices over R is isomorphic to the vector space consisting of 0 and all f (x ) ∈ R [x ] with deg( f ) ≤ 5. Solution. True. (viii) If X and Y are bases of a finite dimensional vector space over a field k , then Y [1V ] X is the identity matrix. Solution. False. (ix) Transposition Matm ×n (C) → Matn ×m (C), given by A → A T , is a nonsingular linear transformation. Solution. True. (x) If V is the vector space of all continuous f : [0, 1] → R, then 1 integration f → 0 f (x ) d x is a linear transformation V → R. Solution. True. 4.33 Let k be a field, let V = k [x ], the polynomial ring viewed as a vector space over k , and let Vn = 1, x , x 2 , . . . , x n . By Exercise 4.8, we know that X n = 1, x , x 2 , . . . , x n is a basis of Vn . (i) Prove that differentiation T : V3 → V3 , defined by T ( f (x )) = f (x ), is a linear transformation, and find the matrix A = X 3 [T ] X3 of differentiation. Solution. Absent. ...
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