Test 4 Review
The test covers sections 4.1 (Kernel and Range only), 4.3, 6.1, 6.3,
and the topics covered in class from 5.1, 5.2, 5.5.
Know all theorems and deﬁnitions presented in class.
Know how to work all
homework problems, especially those not collected for grading.
Deﬁnitions
:
Kernel, range, similar matrices, eigenvalue, eigenvector, characteristic
polynomial (det(
λI

A
)), diagonalizable, matrix exponential, length, distance, angle,
dot product (standard inner product), orthogonal, projection, orthogonal complement,
orthogonal set, orthonormal set
Theorems
:
4.3.1, 6.1.1, 6.3.1, 6.3.2, 5.1.1 (true for
R
n
), 5.5.1 and 5.5.2 (where
V
=
R
n
)
Sample Problems
:
1. Range and Kernel, see test 3 review
2. Let
T
:
R
2
→
R
2
be the linear transformation deﬁned by
T
(
x
1
, x
2
)
T
= (

x
2
,
2
x
1
)
T
.
Consider the two bases
A
=
{
v
1
= (1
,
1)
T
, v
2
= (0
,
1)
T
}
and
B
=
{
w
1
= (1
,
0)
T
, w
2
=
(2
,
1)
T
}
for
R
2
. Find
(a) The matrix A, representing
T
with respect to
A
.
(b) The matrix B, representing
T
with respect to
B
.
(c) The matrix P, such that
B
=
P

1
AP
.
3. Consider the matrix
A
=
4

5
1
1
0

1
0
1

1
(a) Find the eigenvalues and corresponding eigenspaces for
A
.
(b) Find a matrix
P
such that
P

1
AP
is diagonal.
(c) Find
e
A
.
4. For what values of
α
will the following matrix fail to be diagonalizable?
4
6

2

1

1
1
0
0
α
5. page 341 #12 Show that a nonzero nilpotent matrix is defective. Hint: Find the eigen
values of a nilpotent matrix.
1