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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 , ...
View Full Document

## This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

Ask a homework question - tutors are online