Math 2270 Exam 2
-
University of Utah
Fall 2012
Name:
1<
This is a 50 minute exam. Please show all your work, as a worked
problem is required for full points, and partial credit may be rewarded for
some work in the right direction.
1
1. (15 Points) Subspaces
Circle the subsets below that are subspaces of Ii:
6he
set of all vectors
(c) The set of all vectors
)he set of all vectors
(e) The set of all vectors
()
(
a
2
\a3
\\
=
1 +a
a
.
2
with a
3
>
1+a
a
.
2
with a
3
=
0.
with a
3
=
1.
I
()
(
3
with a
a
2
a3
J
2
2. (20 points) For the following matrix, determine the special solutions
for the nulispace, provide a basis for the nulispace, and state the di
mension of the nulispace.
/ 1 2 —2 1 \
A=
3 6 —5 4
1 2 0 3 j
(
10u
A
(
•i1
/
L —Z
5 b vc-I—
zr2f
ro 3
—
I
71
/
t\
0
i
0
1
x1, x
1
f,
x
1
-
$fecl
5o
()
(
cHf
0
/
de%ii’o
3
re
(
3. (20 points) Calculate the rank of the following matrix, provide a basis
for the column space, and state the dimension of the column space:
B—
,4
t
0
o 7
ro I 4C)
7’
-7 I
13
I
/‘t/2Iy rcvL
b
( t’àc1
1
k
-,
\
1
0 —2 1
0 —1 —3 1
—2 —1 1 —1
0
039
o—i
o-0-1-3
0 3
Q
Ii
1
0
3
3
—12
1r&cf towL
-
(
13
(O Oo
\Uo 03-3
o-I
/Oo
7/o( -3
(0
100
-I-i
00 000
\oo o-i
3
/
(
F
0
enh/I a!
1
p
0
(a/mi7
is
4
3,
4. (10 points) For an 8 x 11 matrix A of rank 5 what are:
• dim(C(A))
=
• dim(C(A
))
T
=
• dim(N(A))=
5
jj-
3
• dirn(N(A
))=
T
5
+
0’
II
ç
I
()
N
H
I
II
-
-
If
(I
D
—
r -
—
11
—
Ii
11
II
0
—-
C
—b--
L-
1
cI’j-
I
I
—
co-
—Q
(oo
II
j:QGcJ1
II
H
C
0
CD
C
C
0
Cl)
CD
CD
C
CD
CD
n
(I)
C
c;1
I’J
sil
3 does the following equation have a
6. (10 points) For what value b
solution? (Don’t worry about providing the solution, just give the
.)
3
necessary value of b
2 3\
12
5J
3
3
(\2
i
7
4
I
L
\
1
/
H\131)=(
X2
5bcf
3
rI
3 3
/2\
5).
\63J
/
cfl
i
5
/
I
3
PrY1
o
-
A1s
I
OOQ-7)
7
have
-4-•
cX
—4
-4
U,
Ct
Ct
V
>
1-
(I
ca)
—
H
U
—
I
K
r.-)
Ji
-)
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-
U
H