MATH 2270: Homework 8/Midterm 2 Practice Problems
due October 28, 2015
Instructions: Do the following problems on a separate sheet of paper. Show all of your work.
1. Determine whether or not the following matrix is invertible. Do not try to invert it
(a)

6
9

A=
8
3
4
3
0
−5
0
2
2
−4
6
0
3

4 0
1 0

7 1

0 0
2 0
(b) Find the determinant of A5 .
2. Prove that if det(B 3 ) = 0, then det B = 0.
3. Answer each of the following yes/no questions, and then give an explanation of your reasoning.
(a) Is R3 a subspace of R4 ?
(b) Is P4 a subspace of C(R) = {f : R → R | f is continuous}?
(c) Is the function T : P3 → R4 defined by T (a3 t3 + a2 t2 + a1 t + a0 ) = a3
a vector space isomorphism?
a2
a1
a0
T
4. Show that H = {f ∈ C(R) | f (0) = 0} is a subspace of C(R).
5. Find a linear transformation S : C(R) → R whose kernel is equal to H from the previous
problem.
6. Consider the following matrix:

5 1
3 3

A=
8 4
2 1
6
6
12
3
2
−1
−5
0
 
1 0 1 0
0

−18
 ∼ 0 1 1 0
18  0 0 0 1
0 0 0 0
−3

5
−13

−6 
0
(a) Find a basis for col A.
(b) Find a basis for row A.
(c) Find a basis for ker A.
7. Consider the linear transformation D : P3 → P3 defined by D(p) = p0 +2p. Let B = {1, t, t2 , t3 }
and let C = {1, 1 + t, 1 + t + t2 , 1 + t + t2 + t3 }.
(a) Find the matrix for the linear transformation D with respect to the basis B.
(b) Let p(t) = −2t3 + 3t2 − 10t + 1. Find [p(t)]C .
(c) Find the change of basis matrix P
C←B
and the change of basis matrix P . Hint: It’s
B←C
probably easier to find them both directly than it is to find one and then compute its
inverse.
1
(d) What is the dimension of the image of D? What is the dimension of the kernel of D?
(e) Find a basis for ker D.
8. Is −1
an eigenvector of [ 53 26 ] If so, what is the associated eigenvalue?
1
4
9. Is λ = −3 an eigenvalue of the matrix −1
6 9 ?
10. Find a basis for the eigenspace of the matrix

6
A = 4
2
2
A corresponding to the eigenvalue λ = 2 where

−4 −2
−2 −2
−2 1