189-251B: Algebra 2 Practice Midterm Exam

1. Give the definitions of:

a) A vector space over a field F.

b) Linear independence of vectors.

c) A basis of a vector space.

d) The dimension of a vector space.

2. Let V be a vector space over a field F, and let (v_{1}, …, v_{n}) be a list of vectors in V. Let T : F^{n} → V be the linear transformation defined by

T (x_{1}, …, x_{n}) = x_{1}v_{1} + x_{2}v_{2} + ⋯ + x_{n}v_{n}.

a) Give a necessary and sufficient condition involving the list (v_{1}, …, v_{n}) guaranteeing that T is injective.

b) Give a necessary and sufficient condition involving the list (v_{1}, …, v_{n}) guaranteeing that T is surjective.

3. Let T be a linear transformation on a finite-dimensional vector space and let m(x) be its minimal polynomial. Show that T is invertible if and only if m(0) ≠ 0.

4. Let V denote the vector space of 2 × 2 matrices with entries in F, and let T be the linear transformation that sends a matrix M to its transpose:

T (( \begin{matrix}a & b \\ c & d \end{matrix} )) = ( \begin{matrix}a & c \\ b & d \end{matrix} ) .

Suppose that 2 ≠ 0 in F. Show that T is diagonalisable, by producing a basis of eigenvectors for T. List the eigenvalues of T, and the dimensions of the associated eigenspaces.

Bonus question 5. Let T be a linear transformation and let m(x) ∈ F [x] be its minimal polynomial. Show that g(T) is invertible if and only if gcd(m(x), g(x)) = 1.