189-251A: Algebra 2 Final Exam

Monday, April 23, 2012

This exam has 10 questions, worth 10 points each. Calculators and class notes are not allowed.

1. a) Let F be a field, let S be a finite set and let V be the vector space of functions from S to F. Compute the dimension of V by exhibiting an explicit basis for V.

b) Suppose S is infinite, but countable. Does V then have a countable basis? Justify your answer.

2. Let p(x) be a non-zero polynomial of degree n with coefficients in a field F and let V be the quotient F [x]/(p(x)). Compute the dimension of V by exhibiting an explicit basis for V. (You should include a proof that the set of vectors you come up with is indeed a basis...)

3. a) Let V be the vector space of problem 2, and let T : V ⟶ V be the linear transformation given by T (p(x)) = xp(x).

a) Compute the minimal polynomial of T.

b) Compute the characteristic polynomial of T.

c) Give a necessary and sufficient condition on p(x) for T to be diagonalisable over F.

d) Give a necessary and sufficient conditon on p(x) for T to be invertible.

4. Give an example of a non-zero vector space V and a linear transformation T : V ⟶ V satisfying ker(T) = image(T). Show that such a linear transformation is never diagonalisable.

5. Let V be the vector space of 2 × 2 matrices with entries in the field R of real numbers, and let T : V ⟶ V be the linear transformation given by

T (M) = AMA^{−1}, A = ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) .

Write down a basis for V and the matrix of T relative to this basis.

6. Let T : V ⟶ V be a diagonalisable linear transformation, let λ_{1}, …, λ_{t} be the distinct eigenvalues for T and let

V = ⊕_{i=1}^{t}V_{λ_{i}}

be the associated decomposition of V into a direct sum of eigenspaces. Show that a linear transformation U : V ⟶ V commutes with T if and only if all the eigenspaces V_{λ_{i}} are stable under U. (I.e., if and only if U maps V_{λ_{i}} to itself, for each i.)

7. Define the following terms:

a) The dual space V^{∗} of a vector space V;

b) The dual linear map T^{∗} attached to a linear transformation T : V ⟶ W. Be sure to specify what the domain and target of T^{∗} are, and to write down the formula defining T^{∗}.

c) Show that (T_{1}T_{2})^{∗} = T_{2}^{∗}T_{1}^{∗} for all T_{1} : V ⟶ W and T_{2} : U ⟶ V.

8. State and prove the Cauchy-Scwartz inequality for real inner product spaces.

9. A linear transformation T on a finite-dimensional Hermitan inner product space is said to be skew-adjoint if it satisfies the relation T^{∗} = −T.

a) Show that a skew-adjoint operator is normal.

b) Show that all the eigenvalues of a skew-adjoint operator are purely imaginary.

c) Show that every normal operator T can be written as a sum T_{1} + T_{2} where T_{1} is self-adjoint, T_{2} is skew-adjoint, and T_{1}T_{2} = T_{2}T_{1}.

10. Let V = R^{n} equipped with the standard dot product and resulting distance function, and let W be the hyperplane (i.e., subspace of dimension n − 1) defined by the equation

W = {(x_{1}, …, x_{n}) with x_{1} + ⋯ + x_{n} = 0}.

Show that the vector in W which is closest to the vector (x_{1}, …, x_{n}) is the vector (x_{1} − μ, …, x_{n} − μ), where μ := \frac{x_{1}+···+x_{n}}{n} is the mean of x_{1}, …, x_{n}.