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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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 Fall '11
 KeithCornell
 Linear Algebra, Derivative, Vector Space, Linear map, linear transformation

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