数学代写 - Math 108A HW
2020-12-05
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.