Adv Alegbra HW Solutions 133

Adv Alegbra HW Solutions 133 - 133 x (ii) Prove that...

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Unformatted text preview: 133 x (ii) Prove that integration S : V3 → V4 , defined by S ( f ) = 0 f (t ) dt , is a linear transformation, and find the matrix A = X 4 [ S ] X of in3 tegration. Solution. Absent. 4.34 If σ ∈ Sn and P = Pσ is the corresponding permutation matrix, prove that P −1 = P T . Solution. Absent. 4.35 Let V and W be vector spaces over a field k , and let S , T : V → W be linear transformations. (i) If V and W are finite dimensional, prove that dim(Homk (V , W )) = dim(V ) dim(W ). Solution. Absent. (ii) If X = v1 , . . . , vn is a basis of V , define δ1 , . . . , δn ∈ V ∗ by δi (v j ) = 0 1 if j = i if j = i . Prove that δ1 , . . . , δn is a basis of V ∗ . Solution. Absent. (iii) If dim(V ) = n , prove that dim(V ∗ ) = n , and hence that V ∗ ∼ V . = Solution. Absent. 4.36 (i) If S : V → W is a linear transformation and f ∈ W ∗ , then the S f composite V −→ W −→ k lies in V ∗ . Prove that S ∗ : W ∗ → V ∗ , defined by S ∗ : f → f ◦ S , is a linear transformation. Solution. Absent. (ii) If X = v1 , . . . , vn and Y = w1 , . . . , wm are bases of V and W , respectively, denote the dual bases by X ∗ and Y ∗ . If S : V → W is a linear transformation, prove that the matrix of S ∗ is a transpose: X ∗ [S ∗ Y ∗ = Y [S]X T . Solution. Absent. 4.37 (i) b If A = a d , define det( A) = ad − bc. Given a system of linear c equations Ax = 0 with coefficients in a field, ax + by = p cx + dy = q , ...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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