Algebra 1; MA20008; Sheet 2

            G.C.Smith@bath.ac.uk

18-x-2004

1. Suppose that U and V are vector spaces over the same field F, and that W is a subspace of V. Let α : U → V be a linear map. Show that Y = {x | x ∈ U, α(x) ∈ V } is a subspace of U.

2. Suppose that U and V are vector spaces over the same field F, and that we have linear maps α : U ⟶ V and β : U ⟶ V. Show that Z = {x | x ∈ U, α(x) = β(x)} is a subspace of U.

3. Let ℂ be the complex numbers, viewed as a vector space over ℝ. We have shown that the map φ : ℂ ⟶ ℂ defined by complex conjugation is a linear map. Let n be a natural number, and define θ_{n} to be multiplication by e^{\frac{2πi}{n}}; more formally θ_{n} : ℂ ⟶ ℂ with θ_{n}(z) = e^{\frac{2πi}{n}} z for all z ∈ ℂ.

(a) Show that each map θ_{n} is linear.

(b) How many different maps can you get by composing the maps θ_{4} and θ_{6}? (For example, θ_{4}θ_{4}θ_{6}θ_{6}θ_{4} is one such composition.)

(c) How many different maps can you get by composing the maps θ_{5} and φ?

(d) How many different maps can you get by composing the maps θ_{4}, θ_{6} and φ?

4. Let V we a vector space over a field F. We define a line as follows. Suppose that a, b ∈ V with b ≠ 0. The set

L = {r | r = a + tb, t ∈ F }

is a line. Suppose that U is also a vector space over F and that

α : V ⟶ U

is a linear map. Show that if b ∉ Ker α, then

K = {α(r) | r ∈ L}

is a line. What happens if b = 0?

5. Regard ℝ^{n} as a vector space over ℝ. Define a map μ : ℝ^{n} ⟶ ℝ^{n} by (x_{1}, x_{2}, …, x_{n}) ↦ (x_{2}, x_{3}, …, x_{n}, 0) for all (x_{1}, x_{2}, …, x_{n}) ∈ ℝ^{n}.

(a) Show that μ is a linear map.

(b) Show that μ^{n} is the zero map (μ^{n} denotes the map obtained by composing n copies of μ).

(c) Show that μ^{n−1} is not the zero map.

6. Let V be a vector spaces, and suppose that α and β are both projections onto subspaces of V with suitable kernels. Suppose also that αβ = βα. Show that αβ is a projection.