MA20008 Algebra 1, 2004, Sheet 7

  Geoff Smith, http://www.bath.ac.uk/∼masgcs

1. Let V be a two dimensional vector space over ℝ with basis v_{1}, v_{2}. Let W be a two dimensional vector space over ℝ with basis w_{1}, w_{2}. Suppose that α : V ⟶ W is a linear map which has matrix

( \begin{matrix}1 & 2 \\ 3 & 4 \end{matrix} )

with respect to these bases. Determine the matrix of α with respect to new bases v_{1} + 2v_{2}, v_{1} − v_{2} of V and w_{1} + w_{2}, w_{1} − w_{2} of W.

2. Suppose that A is an m × n matrix with entries in a field F. Show that there are vector spaces V and W over F of dimensions n and m respectively, a linear transformation α : V ⟶ W, and choices of bases for these spaces so that the matrix of α with respect to these bases is A.

3. Let V and W be vector spaces over F of dimensions n and m respectively. Show that there is a choice of bases for V and W such that the matrix of α with respect to these bases is

( \begin{matrix}I_{r} & 0_{r,n−r} \\ 0_{m−r,r} & 0_{m−r,n−r} \end{matrix} ) .

4. Let V = ℝ^{2} viewed as a vector space over ℝ, and thought of as the Euclidean plane. Let ρ_{θ} : V ⟶ V denote the map which is rotation anticlockwise about the origin through θ radians. Choose the standard basis e_{1}, e_{2} for V, where e_{1} = (1, 0) and e_{2} = (0, 1).

(a) Show that ρ_{α} : V ⟶ V is a linear map for every α ∈ ℝ.

(b) Show that ρ_{α} ∘ ρ_{β} = ρ_{α+β} for all α, β ∈ ℝ.

(c) Show that ρ_{α} and ρ_{β} commute for all α, β ∈ ℝ.

(d) Write down the matrix of ρ_{α} with respect to the given basis.

(e) Deduce formulas for cos(α + β) and sin(α + β).

(f) Let c denote complex conjugation. Calculate the matrix of c∘ρ_{θ}∘c and comment on what this means.

5. Suppose that A and B are invertible n × n matrices with real entries. Show that there are invertible n × n real matrices D and E such that DAE = B.

6. Define a relation on n × n matrices with real entries as follows: if A, B ∈ Mat_{n,n}(ℝ), we write A ∼ B if and only if there are invertible P, Q ∈ Mat_{n,n}(ℝ) such that QAP = B.

(a) Show that ∼ is an equivalence relation.

(b) How may equivalence classes are there?