18.700 Problem Set 4

Due in class Friday November 3; late work will not be accepted. Your work on graded problem sets should be entirely your own.

A few of you may notice that the points assigned here add up to 35 instead of 30. You can think of this as extra credit bestowed by a generous professor, or as extra work inflicted by whatever. You can even (in either interpretation) omit one of the seven parts.

1. (20 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 3.

a) Find a basis for V, and therefore compute the dimension of V.

b) If f ∈ V, define a new function T f by

T f = \frac{d^{2}f}{dx^{2}} − (4x^{2} − 10)f.

Prove that T is a linear transformation from V to V. (Hint: the main point is to show that T f ∈ V.)

c) Find the matrix of T in the basis you gave in (a).

d) Find a basis for ker T.

2. (15 points) Let U, V, and W be finite-dimensional vector spaces, and let

T: U → V, S: V → W

be linear transformations.

a) Prove that rank(S ∘ T) ≤ rank S.

b) Prove that rank(S ∘ T) ≤ rank T.

c) Give an example in which it is not true that rank(S ∘ T) is equal to the smaller of rank S and rank T.