Solution
Math 4377/6308 Advanced Linear Algebra
Chapter 2 Review and Solution to Problems
Jiwen He
Department of Mathematics, University of Houston
[email protected]
math.uh.edu/∼jiwenhe/math4377
Jiwen He, University of Houston
Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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Solution
Pb 2.1.15
Recall the definition of P(R) on page 10. Define
Z
T : P(R) → P(R) by T (f (x)) =
x
f (t)dt.
0
Prove that T is linear and one-to-one, but not onto.
Jiwen He, University of Houston
Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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Solution
Pb 2.1.35(a)
Let V be a finite-dimensional vector space and T : V → V be
linear.
(a) Suppose that V = R(T ) + N(T ). Prove that
V = R(T ) ⊕ N(T ).
Be careful to say in each part where finite-dimensionality is used.
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Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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Solution
Pb 2.1.35(a)
(b) Suppose that R(T ) ∩ N(T ) = {0}. Prove that
V = R(T ) ⊕ N(T ).
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Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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Solution
Pb 2.2.12
Let V be a finite-dimensional vector space and T be the projection
on W along W 0 where W and W 0 are subspaces of V such that
V = W ⊕ W 0 . (Recall that a function T : V → V is called the
projection on W along W 0 if, for x = x1 + x2 with x1 ∈ W and
x2 ∈ W 0 , we have T (x) = x1 . Find an ordered basis β for V such
that [T ]β is a diagonal matrix.
Jiwen He, University of Houston
Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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Solution
Pb 2.3.17
Let V be a vector space. Determine all linear transformations
T : V → V such that T = T 2 . Hint: Note that
x = T (x) + (x − T (x)) for every x ∈ V , and show that
V = {y : T (y ) = y } ⊕ N(T ) (see the exercises of Section 1.3).
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Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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Solution
Pb 2.4.17(a)
Let V and W be finite-dimensional vector spaces and T : V → W
be an isomorphism. Let V0 be a subspace of V .
(a) Prove that T (V0 ) is a subspace of W .
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Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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Solution
Pb 2.4.17(b)
(b) Prove that dim(V0 ) = dim(T (V0 )).
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Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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Solution
Pb 2.5.8
Prove the following generalization of Theorem 2.33. Let
T : V → W be a linear transformation from a finite-dimensional
vector space V to a finite-dimensional vector space W . Let β and
β 0 be ordered bases for V , and let γ and γ 0 be ordered bases for
0
W . Then [T ]γβ 0 = P −1 [T ]γβ Q, where Q is the matrix that changes
β 0 -coordinates into β-coordinates and P is the matrix that changes
γ 0 -coordinates into γ-coordinates.
Jiwen He, University of Houston
Math 4377/6308, Advanced Linear Algebra
Spring, 2015
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