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数学代写 - Math 108A HW

时间：2020-12-05

Proof Problems
Your answers to the problems in this section should be proofs, unless otherwise
stated. F is a field, V and W are vector spaces over F.
1) Proposition 14.4 of the Lecture Notes is false when V is an infinite dimensional vector space! Consider the following example:
Let V = {(a1, a2, · · · , an, · · ·) ∈ F∞ | ai = 0 for all but finitely many i} be
the vector space of eventually-zero sequences. Let ei ∈ V be the sequence with a
1 in position i and 0 everywhere else. Define fi : V → F by setting fi(ej ) = δij .
Thus γ∗ = (f1, f2, · · · , fn, · · ·) is the “dual basis” of γ.
a) Prove that γ = (e1, e2, · · · , en, · · ·) is an ordered basis for V .
b) Prove that γ∗
is linearly independent in V ∗ but does not span V ∗
. (In other
words, γ∗
is not actually a basis for V ∗
!)
2) Let V and W be finite-dimensional vector spaces. Let T : V → W be a
linear transformation.
a) Prove that T is surjective if and only if T∗
is injective.
b) Prove that T is injective if and only if T∗
is surjective.
3) Let T : V → W be a linear transformation. Recall the evaluation maps
ev : V → V
∗∗ and ev : W → W∗∗ defined in Theorem 14.11 of the Lecture
Notes. Prove that the following diagram commutes:
1
V W V
∗∗ W∗∗
T
ev ev
T
∗∗
That is, prove that ev ◦ T = T
∗∗ ◦ ev.
4) Let V be a finite-dimensional vector space and let W ⊆ V be a subspace.
Consider (W◦)◦ ⊆ V
∗∗, i.e. (W◦)◦ = {α ∈ V
∗∗ | ∀f ∈ W◦
, α(f) = 0}. Let
ev : V → V
∗∗ be the evaluation map. Prove that ev(W) = (W◦)◦.
5) Let T : V → W be a linear transformation. Prove that the map T : V/ker(T) ∼−→
im(T) given by T(v + ker(T)) = T(v) is well-defined and an isomorphism.
6) Let T : V → V be a linear operator. Let W ⊆ V be a subspace. We say
W is T-invariant if T(w) ∈ W for all w ∈ W.
a) Prove that im(T) and ker(T) are T-invariant.
b) Prove that W is T-invariant if and only if the induced map T : V/W → V/W
given by T(v + W) = T(v) + W is well-defined.
7) Let T : V → V be a linear operator and let W ⊆ V be a T-invariant subspace. Let T |W : W → W denote the restriction of T to W. Prove that
i) ker(T |W ) = ker(T) ∩ W
ii) im(T |W ) = T(W)
8) Let W be a subspace of a vector space V . Let β = {wi}i∈I be a (possibly
infinite) basis of W. Let γ = {vj}j∈J be a disjoint linearly independent set such
that β ∪ γ is a basis for V . Let π : V → V/W be the quotient map.
a) Prove that π(γ) = {vj + W}j∈J is a basis for V/W.
b) Prove that V ∼= W × V/W.
9) Let dim(V ) = n, and let T : V → V be a linear operator. Let W ⊆ V
be a T-invariant subspace of dimension k. Let γ1 = (w1, · · · , wk) be an ordered
basis for W. Let γ2 = (vk+1, · · · , vn) be a disjoint linearly independent set such
that β = (w1, · · · , wk, vk+1, · · · , vn) is an ordered basis of V .
Let T |W : W → W denote the restriction of T to W, and let T : V/W →
V/W be the induced map defined in Problem 6. Let π : V → V/W; recall from
2
Problem 8 that π(γ2) is an ordered basis of V/W.
Write the n × n matrix [T] β β in block form:
[T] β β = A B C D
Here A is a k×k matrix, B is k×(n k), C is (n k)×k and D is (n k)×(n k).
Prove that:
i) A = [T|W ] γ1 γ1
ii) C = 0
iii) D = [T] π(γ2) π(γ2)
Computational Problems
You don’t need to prove your answers to the following questions, but you should
still show your work.
10) Let γ = (e1, e2, e3) be the standard basis for R3
. Let γ∗ = (f1, f2, f3) be
the dual basis of γ. In this problem we shall consider the following question:
What happens to the dual basis if we change only a single vector in γ?
Let β = (e1, e2, e1 + e3). Clearly β is also a basis for R3
. Let β∗ = (g1, g2, g3)
be the dual basis of β. Find a formula for each functional gi and fi
. In other
words, find an expression for gi a1 a2 a3 in terms of a1, a2, and a3.
Note that β differs from γ only in the third vector. Is g1 = f1 and g2 = f2?
Explain why or why not.