Linear Algebra II (pdf, 500 kB)

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6.15. Proposition. Let V and W be vector spaces, f : V → W a linear map.
Then the following diagram commutes.
αV
V
f
W
αW
f ⊤⊤
∗∗ V
∗∗ W
Proof. We have to show that f ⊤⊤ ◦ αV = αW ◦ f. So let v ∈ V and w ∗ ∈ W ∗ .
Then
f ⊤⊤ αV (v) (w ∗ ) = (αV (v) ◦ f ⊤ )(w ∗ ) = αV (v) f ⊤ (w ∗ )
= αV (v)(w ∗ ◦ f) = (w ∗ ◦ f)(v)
= w ∗ f(v) ∗
= αW f(v) (w ) .
6.16. Proposition. Let V be a vector space. Then we have α⊤ V
If V is finite-dimensional, then α⊤ V
Proof. Let v ∗ ∈ V ∗ , v ∈ V. Then
α ⊤ V
so α ⊤ V
= α−1
V ∗.
◦ αV ∗ = idV ∗.
αV ∗(v∗ ) (v) = αV ∗(v∗
) ◦ αV (v) = αV ∗(v∗ ) αV (v) = αV (v) (v ∗ ) = v ∗ (v) ,
αV ∗(v∗ ) = v ∗ , and α ⊤ V
◦ αV ∗ = idV ∗.
If dim V < ∞, then dim V ∗ = dim V < ∞, and αV ∗ is an isomorphism; the
relation we have shown then implies that α ⊤ V
= α−1
V
∗.
6.17. Corollary. Let V and W be finite-dimensional vector spaces. Then
is an isomorphism.
Hom(V, W ) ∋ f ↦−→ f ⊤ ∈ Hom(W ∗ , V ∗ )
Proof. By the observations made in Def. 6.11, the map is linear. We have another
map
Hom(W ∗ , V ∗ ) ∋ φ ↦−→ α −1
W ◦ φ⊤ ◦ αV ∈ Hom(V, W ) ,
and by Prop. 6.15 and Prop. 6.16, the two maps are inverses of each other:
and
α −1
W ◦ f ⊤⊤ ◦ αV = f
(α −1
W ◦ φ⊤ ◦ αV ) ⊤ = α ⊤ V ◦ φ ⊤⊤ ◦ (α ⊤ W ) −1 = α −1
V ∗ ◦ φ⊤⊤ ◦ αW ∗ = φ .
Next, we study how subspaces relate to dualization.
6.18. Definition. Let V be a vector space and S ⊂ V a subset. Then
is called the annihilator of S.
S ◦ = {v ∗ ∈ V ∗ : v ∗ (v) = 0 for all v ∈ S} ⊂ V ∗
S ◦ is a linear subspace of V ∗ , since we can write
S ◦ =
ker αV (v) .
v∈S
Trivial examples are {0V } ◦ = V ∗ and V ◦ = {0V ∗}.