Algebra 1; MA20008; Sheet 3

            G.C.Smith@bath.ac.uk

25-x-2004

1. Suppose that V is a vector space over F, and that S ⊆ V. Let \overline{S} be the intersection of those subspaces of V which contain the subset S, or put formally

\overline{S} = ⋂\{U | U ≤ V, S ⊆ U\}.

Show that \overline{S} = ⟨S⟩.

2. Let V be a vector space over F and v_{1}, v_{2}, …, v_{n} ∈ V. Suppose that whenever θ_{1}, …, θ_{n}, ψ_{1}, …, ψ_{n} ∈ F and ∑_{i=1}^{n} θ_{i}v_{i} = ∑_{i=i}^{n} ψ_{i}v_{i}, then necessarily θ_{i} = ψ_{i} for each i, 1 ≤ i ≤ n. Show that v_{1}, v_{2}, …, v_{n} ∈ V is linearly independent.

3. Consider V = ℂ as a vector space over ℚ.

(a) Show that 1, \sqrt{2}, \sqrt{3} are linearly independent.

(b) Let α = e^{\frac{πi}{3}} . Which lists of the form 1, α, …, α^{n} are linearly independent? Justify your answer.

(c) Suppose that 1, β, β^{2}, …, β^{n} are linearly independent. Show that 1, (β + 1), (β + 1)^{2}, …, (β + 1)^{n} are linearly independent.

4. Suppose that X and Y are both linearly independent subsets of V. Does it follow that X ∩Y is linearly independent? What about X ∪Y?

5. Suppose that V = U ⊕ W. We are given a sets of vectors X ⊆ U and Y ⊆ W. Is X ∪ Y necessarily a linearly independent set of vectors?

6. Suppose that v_{1}, v_{2}, …, v_{n} is a linearly independent list of vectors in the vector space V. We are given w ∈ V. Does it follow that

v_{1} + w, v_{2} + w, …, v_{n} + w

are linearly independent?

7. Suppose that v_{1}, v_{2}, …, v_{n} is a linearly dependent list of vectors in the vector space V. We are given w ∈ V. Does it follow that

v_{1} + w, v_{2} + w, …, v_{n} + w

is linearly dependent?

8. Let V = ℝ^{3} viewed as vector space over ℝ. Let v_{1}, …, v_{8} be the position vectors of the vertices of a cube.

(a) Let

A = {∑_{i} λ_{i}v_{i} | 0 ≤ λ_{i} ≤ 1 for all i, ∑_{i} λ_{i} = 1} .

Describe the set A, viewed as a collection of position vectors, geometrically.

(b) Let

B = {∑_{i} λ_{i}v_{i} | λ_{i} ≥ 0 for all i, } .

Under what circumstances is B = ℝ^{3}? Under what circumstances is B a closed half space (i.e. one side of a plane and all the points on that plane)? What other shapes can arise?