FINAL FOR ADVANCED LINEAR ALGEBRA
Date: Wednesday, January 17, 2001 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1.
(i) [5%] Give an example of an algebra over C of dimension 5.
(ii) [5%] Let V be a three-dimensional vector space over R. Give an example of a nonzero skew- symmetric bilinear form on V .
(iii) [5%] Give an example of a degree two representation of the group Z3.
2. [10%] Let V be an inner product space over R. Suppose that S, T are subspaces of V such that V is the orthogonal direct sum of S and T . Show that the radical of V is the orthogonal direct sum of the radical of S and the radical of T i.e., prove that rad(V ) = rad(S) ⊥ rad(T ) if V = S ⊥ T .
3. [10%] Let V, W be vector spaces over a field F . Show that V ⊗ W and W ⊗ V are isomorphic.
4. [10%] Find an example of a bilinear map τ : V × V → W whose image Im(τ ) = { τ (u, v) | u, v ∈ V } is not a subspace of W .
5. Let R be a commutative ring with identity.
(i) [5%] Show that any two nonzero elements in R are not linearly independent.
(ii) [5%] Using (i) conclude that an ideal I of R is a free R-module if and only if I is generated by an element of R that is not a zero divisor.
6. [10%] Suppose that V, W are vector spaces over a field F . Let f : V × V → W be a map from V × V to W . Suppose that f is both linear and bilinear. Show that f is a zero map.
7. [10%] Let ρ denote the regular representation of the group Z5. Decompose ρ as a direct sum of irreducible representations.
8. [10%] Let ρ1: G1 → GL(V1) and ρ2: G2 → GL(V2) be representations of finite groups G1, G2
respectively. Define ρ1⊗ ρ2: G1⊕ G2→ GL(V1⊗ V2) by
(ρ1⊗ ρ2)(g1, g2)
ÃX
finite
vi⊗ wi
!
= X
finite
ρ1(g1)(vi) ⊗ ρ2(g2)(wi)
for g1∈ G1, g2∈ G2, vi ∈ V1 and wi∈ V2. Check that ρ1⊗ ρ2 is a representation of G1⊕ G2.
9. Let G be the group S3 i.e., G is a non-abelian group of order 6 with elements {1, a, a2, b, ba, ba2} and the relations a3 = 1, b2 = 1 and ab = ba2. Let ρ be the regular representation of G, ρ1 be the trivial representation of G. Define ρ2: G → C∗ by defining ρ2(a) = 1 and ρ2(b) = −1.
(i) [5%] Check that ρ2 can be made into a representation of G.
(ii) [5%] We know that there is a representation ρ3of G such that ρ = ρ1⊕ ρ2⊕ ρ3. Is ρ3irreducible?
Why or why not?
(iii) [5%] Find ρ3(a).
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