Algebra 1; MA20008; Sheet 4

            G.C.Smith@bath.ac.uk

2-xi-2004

1. Suppose that n is a natural number or 0, and F is a field. Show that there is a vector space over F of dimension n.

2. Suppose that V is a vector space with subspaces U and W both of dimension n < ∞. Does it follow that V is finite dimensional? Does it follow that V has dimension n? Does it follow that U = W? In each case you should supply a reason for your answer.

3. Let V be a vector space of dimension n. Suppose that V_{0}, V_{1}, …, V_{m} are subspaces of V with

V_{0} ≤ V_{1} ≤ ⋯ ≤ V_{m}.

(a) Suppose that m > n. Show that there is i ∈ {1, 2, …, m} such that V_{i} = V_{i−1}.

(b) Suppose that m ≤ n. Show that it may be that the spaces V_{0}, V_{1}, …, V_{m} are distinct.

4. Suppose that α : U ⟶ W is a linear map between vector spaces over the same field. Let x_{1}, x_{2}, …, x_{n} be vectors in U.

(a) Suppose that U = ⟨x_{1}, x_{2}, …, x_{n}⟩ and α is surjective. Prove that W = ⟨α(x_{1}), α(x_{2}), …, α(x_{n})⟩.

(b) Suppose that x_{1}, x_{2}, …, x_{n} are linearly independent and α is injective. Show that α(x_{1}), α(x_{2}), …, α(x_{n})⟩ are linearly independent.

5. Let ζ = e^{\frac{2πi}{5}} ∈ ℂ.

(a) Suppose that we view ℂ as a vector space over ℚ. Show that 1, ζ, ζ^{2}, ζ^{3} are linearly independent.

(b) Suppose that we view ℂ as a vector space over ℝ. Show that 1, ζ, ζ^{2}, ζ^{3} are linearly dependent.

6. Let V be a vector space with subspaces U, W such that U and W are both finite dimensional. Let u_{1}, u_{2}, …, u_{m} be a basis of U and w_{1}, w_{2}, …, w_{n} be a basis of W.

(a) Show that U + W is finite dimensional.

(b) Show that u_{1}, u_{2}, …, u_{m}, w_{1}, w_{2}, …, w_{n} need not be a basis of U + W.

(c) Suppose that U + W = U ⊕ W. Show that

u_{1}, u_{2}, …, u_{m}, w_{1}, w_{2}, …, w_{n}

is a basis of U + W.

(d) Suppose that u_{1}, u_{2}, …, u_{m}, w_{1}, w_{2}, …, w_{n} is a basis of U + W. Show that U + W = U ⊕ W.

7. Let I be a set. Let V be the set of real valued functions on I; more

formally

V = {f|f : I ⟶ ℝ} .

Define addition on V by (f + h)(x) := f(x) + g(x) for all x ∈ I. If λ ∈ ℝ and f ∈ V we define λ · f ∈ V by (λ · f)(x) = (λ)(f(x)) where the final multiplication is just the product (in ℝ).

(a) Check that V is now a vector space over ℝ.

(b) For each i ∈ I, define a function δ_{i} ∈ V where δ_{i}(x) = δ_{i,x} (Krönecker delta). Thus δ_{i}(i) = 1 and δ_{i}(x) = 0 if x ≠ i. Show that the vectors δ_{i} are linearly independent.

(c) Let W = ⟨δ_{i} : i ∈ I⟩ be the span of all the δ_{i}. Show that the vectors δ_{i} form a basis of W (in that they are a linearly independent spanning set for W).

(d) Show that W = V if and only if I is finite.

(e) Give an explicit example of a vector space with an clearly describable uncountable basis (no set theoretic metaphysics allowed).