Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 304 — Problem Set 9
Issued: 11.08
Due: 11.16
9.1. Determine whether the following are linear transformations from the
space of differentiable functions to the space of functions.
(a) L(f (x)) = xf 0 (x);
(b) L(f (x)) = x2 + f (x).
9.2. Find the standard matrix representation (i.e., the representation in
the standard bases) for the following linear operators:
(a) The linear operator R2 −→ R2 that rotates each vector ~x in R2
by 45◦ in the clockwise direction.
(b) The linear operator R2 −→ R2 that doubles the length of ~x and
then rotates it by 30◦ in the counterclockwise direction.

 

2x3
x1
(c) The operator R3 −→ R3 given by the formula L  x2  =  x2 + 3x1 .
2x1 − x3
x3
9.3. Let S be the subspace of C[a, b] spanned by ex , xex , and x2 ex . Let
D(f ) = f 0 be the differentiation operator on S. Find the matrix representation of D with respect to the basis {ex , xex , x2 ex }.
9.4. Let L be the linear operator on R3 defined by L(~x) = A~x, where


3 −1 −2
0 −2 
A= 2
2 −1 −1
and let


1
~v1 =  1  ,
1


1
~v2 =  2  ,
0

0
~v3 =  −2  .
1
Find the matrix of L in the basis {~v1 , ~v2 , ~v3 }.



1 2 −1
2 .
9.5. Find characteristic polynomial of the matrix A =  0 1
0 2
1