MA20008 Algebra 1, 2004, Sheet 6

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

1. Let V be a vector space of dimension n. Suppose that α : V → V is a linear map. Show that the following are equivalent.

(a) α is injective.

(b) α is bijective.

(c) α is surjective.

(d) There is a basis v_{1}, …, v_{n} of V such that α(v_{1}), …, α(v_{n}) is a basis of V.

(e) For every basis v_{1}, …, v_{n} of V, the vectors α(v_{1}), …, α(v_{n}) also form a basis of V.

Hint: the rank-nullity theorem may be useful in places.

2. Suppose that

X = ( \begin{matrix}A & B \\ 0 & C \end{matrix} )

is a 2r by 2r matrix built from the four r by r matrices A, B, C and the zero matrix 0. Suppose that X has an inverse matrix. Describe that matrix in terms of A, B, C and 0.

3. The matrix

F = ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} )

has entries in the Field F_{7}, the integers modulo 7. Calculate

(a) F^{2}.

(b) F^{5}.

(c) F^{1000}.

∑_{i=0}^{999} F^{i}

where F^{0} denotes the identity matrix.

4. The matrix

F = ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} )

has entries in ℚ, Let I denote the 2 by 2 identity matrix. Show that I and F are linearly independent but that I, F and F^{2} are linearly dependent elements of the vector space of 2 by 2 matrices with rational entries (with scalars in ℚ).

5. Suppose that α, β : V ⟶ V are a pair of commuting linear maps. (a) Prove that both Im α and Ker α are β-invariant spaces.

(b) Prove that Im α + Im β is both α-invariant and β-invariant.

(c) Prove that Im α ∩ Im β is both α-invariant and β-invariant.