18.700 Problem Set 5

Due in class Monday November 20; late work will not be accepted. Your work on graded problem sets should be entirely your own.

1. (8 points) Let V be the real vector space consisting of all functions on the real line of the form p(x)e^{−x^{2}}, with p a polynomial of degree at most 5. If f ∈ V, define a new function T f by

T f = \frac{d^{2}f}{dx^{2}} − 4x^{2}f.

You more or less showed on the last problem set that T is a linear operator on V. Find all the eigenvalues of T, and decide whether T is diagonalizable.

2. (7 points) Suppose A is a 3 × 3 real matrix. Prove that A has at least one (real) eigenvector.

3. (8 points) Suppose the ground field is the complex numbers, θ is a real number, and

R = ( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ) .

Find all the eigenvalues and eigenvectors of R, and show how to change bases to diagonalize R.

4. (7 points) Suppose the ground field is the real numbers, θ is a real number, and

R = ( \begin{matrix}cos θ & sin θ \\ sin θ & − cos θ \end{matrix} ) .

Find all the eigenvalues and eigenvectors of R, and show how to change bases to diagonalize R.