Homework 15, due 12/7/2015
1. Artin 8.1.1. What fields can you take in place of R and have the result
still hold?
2. Artin 8.3.2
3. Artin 8.3.4
4. Artin 8.4.2
5. Artin 8.4.5
6. Let h, i : V × V → F be a non-degenerate symmetric bilinear form on a
finite-dimensional vector space V over a field F . This exercise will show in
steps that if the restriction of h, i to a subspace W of V is non-degenerate,
then V = W ⊕ W ⊥ .
(a) Show W ∩ W ⊥ = (0).
(b) Show the map Λ : V → HomF (V, F ) (recall this Hom space is denoted
V ∗ ) given by v 7→ Λv , Λv (w) = hv, wi, is an isomorphism.
(c) What is the image of W ⊥ under Λ?
(d) Exhibit an isomorphism
∼
(V /W )∗ −
→ {λ ∈ V ∗ : λ|W = 0}.
(e) Deduce that dim W + dim W ⊥ = dim V , hence (using part (a)) that
V = W ⊕ W ⊥.
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