Advanced Linear Algebra Math 424/524

Assignment No. 4

Due December 2, 2002

1. Let V be a n-dimensional vector space over a field F, v be a basis of V, and w the dual basis of V^{∗}. Let v′ be another basis of V and w′ the basis of V^{∗} dual to it. If the change of basis between v and v′, regarded as rows of vectors, is given by v′ = vA for an n × n matrix A and if also for V^{∗} one has w′ = wB, how is the matrix B related to the matrix A?

2. Let f_{j} : V_{j} ⟶ W_{j} for j = 1, 2 be linear maps.

(a) Explain briefly why there is a unique linear map

f_{1} ⊗ f_{2} : V_{1} ⊗ V_{2} ⟶ W_{1} ⊗ W_{2}

for which

(f_{1} ⊗ f_{2})(v_{1} ⊗ v_{2}) = f_{1}(v_{1}) ⊗ f_{2}(v_{2}) whenever v_{1} ∈ V_{1}, v_{2} ∈ V_{2} .

(b) If for chosen bases in V_{j}, W_{j}, the linear map f_{j} has the matrix

( \begin{matrix}a_{j} & b_{j} \\ c_{j} & d_{j} \end{matrix} ) for j = 1, 2 ,

what is the matrix of f_{1} ⊗ f_{2} with respect to suitable orderings of the product bases for V_{1} ⊗ V_{2} and W_{1} ⊗ W_{2}?

3. Let R denote the field of real numbers. What more familiar description may be used to describe the dth symmetric power S^{d}((R^{2})^{∗}) of the dual space of the real plane R^{2} to a student who has completed the calculus sequence?

4. Let Q denote the field of rational numbers, P (t) the polynomial t^{5} + 5t^{4} + 6t^{3} + 7t^{2} + 8t + 9

in Q[t], and let

V = Q[t]/P (t)Q[t] .

(a) Prove that V has dimension 5 over Q by exhibiting a basis of V.

(b) If π denotes the quotient map from Q[t] to V, and τ denotes the linear endomorphism of Q[t] defined by τ(F (t)) = tF (t) for F ∈ Q[t], explain briefly why there is a linear endomorphism φ of V such that φ ∘ π = π ∘ τ.

(c) Find the matrix of φ with respect to the basis of V determined in response to the first part of this exercise.

(d) What is the characteristic polynomial of the matrix found in the previous part of this exercise?

5. Let V and W be vector spaces over a field F. There is a natural linear map

ρ : V ⊗ W ⟶ Hom(V^{∗}, W)

that is uniquely determined by specifying for v ∈ V, w ∈ W

ρ(v ⊗ w) = {f ↦ f(v)w for f ∈ V^{∗}} .

Prove that ρ is an isomorphism if V and W are finite-dimensional.