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MATH227 Quiz 2 (Fall 2006)
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Let
A
= [
a
1
a
2
a
3
] =
2
0
6

1
8
5
1

2
1
,
b
=
10
3
3
, and
W
be the set of all linear
combinations of the columns of
A
(i.e.,
W
= Span
{
a
1
,
a
2
,
a
3
}
).
(a) Show that
0
and
a
2
are in
W
.
(4 points)
(b) Is the vector
b
in
W
? How about the vector
a
1
+
b
?
(6 points)
(a) Since
0
= 0
·
a
1
+ 0
·
a
2
+ 0
·
a
3
and
a
2
= 0
·
a
1
+ 1
·
a
2
+ 0
·
a
3
,
the vectors
0
and
a
2
are in
W
= Span
{
a
1
,
a
2
,
a
3
}
.
(b) Consider the augmented matrix
[
a
1
a
2
a
3

b
]
=
2
0
6

10

1
8
5

3
1

2
1

3
R
2
+
1
2
R
1
R
3

1
2
R
1
∼
2
0
6

10
0
8
8

8
0

2

2


2
R
3
+
1
4
R
2
∼
2
0
6

10
0
8
8

8
0
0
0

0
.
We see from the above row echelon form that the system of linear equations with
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Unformatted text preview: augmented matrix [ a 1 a 2 a 3  b ] is consistent. Then, it is equivalent to say that b is a linear combination of a 1 , a 2 , a 3 . That is, b is in W . Now, as b is in W , b = c 1 a 1 + c 2 a 2 + c 3 a 3 for some scalars c 1 , c 2 , c 3 . Then a 1 + b = ( c 1 + 1) a 1 + c 2 a 2 + c 3 a 3 . Therefore, a 1 + b is also in W . 1...
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This note was uploaded on 04/23/2008 for the course MATH 227 taught by Professor Sze during the Fall '06 term at UConn.
 Fall '06
 Sze
 Math

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