Math 3A (44351)
Quiz 3
Problem 1.
Let
A
1
=
1
0
3
4
, A
2
=
4
0

2
3
, A
3
=
0
0
1
2
Show that
T
=
7
0
0
1
∈
Span(
A
1
, A
2
, A
3
)
.
Do
{
A
1
, A
2
, A
3
}
span
all of
R
2
x
2
? Explain.
Solution:
3
A
1
+
A
2

7
A
3
=
3
0
9
12
+
4
0

2
3

0
0
7
14
=
7
0
0
1
and so
T
is a linear combination of
A
1
, A
2
, A
3
, and thus
T
∈
Span(
A
1
, A
2
, A
3
).
Since all three vectors have a topright component of 0, it is clear
that the vector
0
1
0
0
, which is in
R
2
x
2
, is not in Span(
A
1
, A
2
, A
3
),
so no, they do not span all of
R
2
x
2
Problem 2.
Suppose that
{
v
1
, v
2
, . . . , v
k
}
, with
k >
2, is a set of lin
early independent vectors in
R
n
. Prove that the set
{
v
2
, . . . , v
k
}
must
also be linearly independent.
Solution: Suppose that
c
2
v
2
+
. . .
+
c
k
v
k
= 0 for some set of constants
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 Spring '10
 PANTANO
 Math, Vector Space, ck vk

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