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Math 1b Practical — A theorem on linear dependence
January 12, 2011
Theorem.
If vectors
v
1
,
v
2
,...,
v
k
are linear combinations of vectors
u
1
,
u
2
u
r
,and
if
k>r
,then
v
1
,
v
2
v
k
are linearly dependent.
We illustrate with an example. Suppose
k
=4and
r
=3,andthat
3
u
1
+
u
2
+4
u
3
=
v
1
−
2
u
1
+3
u
3
=
v
2
−
4
u
1
+2
u
2
−
u
3
=
v
3
5
u
1
−
u
2
u
3
=
v
4
We may use row operations to get new and equivalent sytems of equations, e.g. we may
replace equation 1 by the sum of equation 1 and 7 times equation 2. As usual, we can do
this with the matrix of coeFcients alone. So start with
M
=
⎛
⎜
⎝
314
v
1
−
20
3
v
2
−
42
−
1
v
3
5
−
12
v
4
⎞
⎟
⎠
or
⎛
⎜
⎝
3
1
4 1000
−
30
1
0
0
−
−
10010
5
−
120
0
0
1
⎞
⎟
⎠
and use pivot operations in the Frst three columns to make the inital 4
×
3 matrix basic.
(We may put linear combinations of the symbols
v
i
in the last column, or we may Fnd it
more convenient to add four columns for the coeﬃcients of the
v
i
’s.)
We obtain
⎛
⎜
⎝
100
(
2
v
1
−
3
v
2
−
v
3
)
/
16
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 Winter '08
 Aschbacher
 Vectors

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