MAS 3105
Oct 17, 2008
Quiz 4 and Key
Prof. S. Hudson
1) The matrix
A
is row equivalent to
U
. a) Find a basis for
N
(
A
), and explain briefly how
you know it is a basis. b) Do the same for Col
A
.
A
=
1
2

1
1
2
4

3
0
1
2
1
5
U
=
1
2
0
3
0
0
1
2
0
0
0
0
2) Are the following sets subspaces of
R
3
? Explain each answer briefly.
a)
S
=
{
(
x
1
, x
2
, x
3
)
T

x
1
=
x
2
and
x
3
= 0
}
b)
S
=
{
(
x
1
, x
2
, x
3
)
T

x
1
=
x
2
or
x
1
=
x
3
}
3) Choose ONE of these to prove. You can answer on the back.
a) If dim(V)=
n >
0 and
B
=
{
v
1
, v
2
, . . . , v
n
}
is L.I., then
B
is a basis of V.
b) If
U
and
V
are subspaces of
W
then so is
U
∩
V
.
c) If
B
=
{
v
1
, v
2
, . . . , v
n
} ⊂
V
is L.I., then any subset of
B
is also L.I.
Remarks and Answers:
I don’t have the scores from the grader yet, but he estimates
that the average was about 35/60. If so, an approx scale is A’s 4560, B’s 3944, C’s 3338,
D’s 2732.
He said some students couldn’t write proofs, using the definition of LI for example.
You had graded HW on that, so I suppose you know whether this is a problem for you. If
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 Spring '09
 JULIANEDWARDS
 Trigraph, li, col, ck vk, Prof. S. Hudson

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