Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Summer 2015
Math 304 — Problem Set 4
Issued: 7.28
Due: training
4.1. Find a basis of the column space, row space, and null space of the
matrix


1 3 1 3 4
 0 0 1 1 1 


 0 0 2 2 2 .
0 0 3 3 3


x1
4.2. Is the set of vectors  x2  such that x1 − x2 + 2x3 = 0 a subspace
x3
3
of R ? If yes, what is its dimension?
4.3. Is the set of polynomials p(x) such that p0 (x) = p(0) a subspace of the
space of polynomials of degree at most 3?
4.4. What is the dimension of the subspace of R3 spanned by the vectors
    

1
2
4
 2 , 5 , 9 .
3
7
13


5
4.5. Does the vector  10  belong to the column space of the matrix
5


1 2
 2 4 ?
1 2
4.6. Find the transition matrix from the basis 1, x − 1, (x − 1)2 to the basis
1, x + 1, (x + 1)2 of the space of polynomials of degree at most 2.
4.7. 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).
4.8. For each of the following linear transformations L mapping R3 into R2
or R3 , find the matrix A such that L(~x) = A~x for every ~x in R3 .


x1
x1 + x2


(a) L x2
=
.
0
x3


x1
x
−
x
2
1
.
(b) L  x2  =
x3 + x2
x3

 

x1
2x3
(c) L  x2  =  x2 + 3x1 .
x3
2x1 − x3
4.9. Find the standard matrix representation (i.e., the representation in
the standard bases) for the following linear operators on R2 :
(a) The linear operator that rotates each vector ~x in R2 by 45◦ in the
clockwise direction.
(b) The linear operator that doubles the length of ~x and then rotates
it by 30◦ in the counterclockwise direction.
4.10. 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 }.
4.11. Determine the kernel and range of the linear operator

 

x1
x1
L  x2  =  x3  .
x3
0
4.12. 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 }.