Math 304
Examination 2
Linear Algebra
Summer 2006
Write your
name
:
Answer Key
(2 points).
In
problems 1–5
, circle the correct answer. (5 points each)
1. On the vector space of polynomials, differentiation is a linear operator.
True
False
Solution.
True: the derivative of a sum is the sum of the derivatives,
and the derivative of a scalar times a polynomial is the scalar times the
derivative of the polynomial.
2. If the linear system
A
x
=
b
is consistent, then the vector
b
must be in
the space
N
(
A
)
⊥
.
True
False
Solution.
False. If
A
is an
m
×
n
matrix, then
b
is in
R
m
and
N
(
A
)
⊥
is a subspace of
R
n
, so the statement does not even make sense when
m
negationslash
=
n
. What is true is that
b
must be in the range
R
(
A
), and that
space is equal to the space
N
(
A
T
)
⊥
(not
N
(
A
)
⊥
).
3. The matrix
parenleftbigg
0
−
1
−
1
0
parenrightbigg
is the matrix representation (with respect to
the standard basis) of the linear operator that reflects each vector
x
in
R
2
about the
x
2
axis and then rotates it 90
◦
in the counterclockwise
direction.
True
False
Solution.
True: the image of the first basis vector
parenleftbigg
1
0
parenrightbigg
is
parenleftbigg
0
−
1
parenrightbigg
, the
first column of the matrix, and the image of the second basis vector
parenleftbigg
0
1
parenrightbigg
is
parenleftbigg
−
1
0
parenrightbigg
, the second column of the matrix.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 HOBBS
 Linear Algebra, Polynomials, Vector Space, Dr. Boas, True False Solution

Click to edit the document details