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Math 43, Fall 2007
B. Dodson
Week 5:
Spanning, Independence and Bases
Is the vector
±
b
= [3
,
3
,
4] a linear combination of
±v
1
= [1
,

1
,
2]
,±v
2
= [2
,
1
,
3]?
Solution
We have
3
3
4
=
c
1
1

1
2
+
c
2
2
1
3
=
1
2

1
1
2
3
•
c
1
c
2
‚
,
where we use matrix notation from 3.1.
We can easily see the system of equations directly, without using this
matrix product, but writing the vector equation as the
matrix equation
A±
c
=
±
b
gives the augmented matrix of the system
directly as [
A

±
b
]
.
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Observe that this is the linear system
with augmented matrix [
±v
1
±v
2

±
b
] =
1
2

3

1
1

3
2
3

4
.
Row reduction gives
1
2

3
0
3

6
0

1
 
2
.
We see that this is a consistent system. (why?) So YES,
±
b
is a linear combination. We are often asked to continue by
ﬁnding an explicit linear combination, solving for
c
1
, c
2
.
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This note was uploaded on 02/29/2008 for the course MATH 43 taught by Professor Dodson during the Spring '08 term at Lehigh University .
 Spring '08
 Dodson
 Math

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