189-251B: Algebra 2 Assignment 2

Due: Wednesday, January 22

1. Let C be the field of complex numbers and let V be the C-vector space C^{3}. Find the coordinates of the vector (1, 0, 1) in the basis (v_{1}, v_{2}, v_{3}), where

v_{1} = (2i, 1, 0), v_{2} = (2, −1, 1), v_{3} = (0, 1 + i, 1 − i).

2. Let V denote the Q-vector space of 2 × 2 matrices with entries in Q. Let

A be the matrix ( \begin{matrix}2 & −5 \\ 3 & 17 \end{matrix} ). Show that the function

T : V ⟶ V defined by T (X) = AX − XA

is a linear transformation from V to V. Choose a basis for V, and write down the matrix of T with respect to your basis.

3. Let V be a finite-dimensional vector space.

a. Show that any linear transformation T : V ⟶ V is injective if and only if it is surjective.

b. Show that neither property implies the other, if the finite-dimensionality assumption on V is dropped.

4. Let V and W be finite-dimensional vector spaces, of dimension n and m respectively, over the field Z_{2}. How many functions are there from V to W? How many linear transformations? What are the dimensions of these spaces of functions, and linear transformations, respectively?

5. Let V be the set of real-valued sequences (a_{n})_{n≥0} satisfying

a_{n+1} = a_{n} + a_{n−1} for all n ≥ 1.

a. Show that V, equipped with the usual internal addition and scalar multiplication on sequences, is a vector space over R.

b. What is the dimension of V over R?

c. A geometric progression is a sequence of the form a_{n} = α^{n}, for some α ∈ R. Show that V has a basis consisting of geometric progressions, by producing such a basis.

d. The Fibonacci sequence is the unique sequence (a_{n}) ∈ V satisfying

a_{0} = a_{1} = 1, so that a_{2} = 2, a_{3} = 3, a_{4} = 5, a_{5} = 8, a_{6} = 13, etc.

Express this sequence as a linear combination of the basis vectors obtained in part c. Deduce from this a closed form expression for the nth term of the Fibonacci sequence.