T1: Lectures 11-15

Question 1

Given a vector space V, define the dual space V*

Answer 1

V* = Hom_G(V, C)

Question 2

Given two vector spaces V and W, define V⊗ ̃W

Answer 2

The span of all finite linear combinations of elements of v ⊗ w

Question 3

Define the subset U_1 of V⊗ ̃V

Answer 3

Span of v_1 ⊗ v_2 + v_2 ⊗ v_1

Question 4

Define V⊗W

Answer 4

V⊗W = V⊗ ̃W/U

Question 5

Given V with basis {v_i} and W with basis {w_j} what is the basis of V⊗W?

Answer 5

{v_i⊗w_j}

Question 6

Given an n dimensional vector space V, what is the dimension of sym^2(V)?

Answer 6

n(n+1)/2

Question 7

Given a vector space V and subspace U_1 = span{v_1 ⊗ v_2 - v_2 ⊗ v_1} define sym^2 (V)

Answer 7

sym^2 (V) = V⊗V/U_1

Question 8

Given a vector space V and subspace U_2 = span{v_1 ⊗ v_2 + v_2 ⊗ v_1} define Λ^2 (V)

Answer 8

Λ^2 (V) = V⊗V/U_2

Question 9

Given an n dimensional vector space V, what is the dimension of Λ^2(V)?

Answer 9

n(n-1)/2

Question 10

Define the wedge uΛv

Answer 10

uΛv := u⊕v+U_2

Question 11

What rule (additional to sym^2) is the alternating square subject to?

Answer 11

Elements behave as vΛv’ = -v’Λv

Question 12

Give the character of sym^2 (a)

Answer 12

1/2 (a(g)^2 + a(g^2))

Question 13

Give the character of the alternating square

Answer 13

1/2 (a(g)^2 - a(g^2))