Unformatted text preview: zero
vector is linearly dependent.
Proof: Renumber the vectors so that v 1 ____. Then
____v 1 _____v 2 _____v p 0
which shows that S is linearly ________________.
4. A Set Containing Too Many Vectors
Theorem 8
If a set contains more vectors than there are entries in each
vector, then the set is linearly dependent. I.e. any set
v 1 , v 2 , , v p in R n is linearly dependent if p n.
Outline of Proof:
A v1 v2 vp is n p Suppose p n.
Í Ax 0 has more variables than equations
Í Ax 0 has nontrivial solutions
Ícolumns of A are linearly dependent 9 EXAMPLE
With the least amount of work possible, decide
which of the following sets of vectors are linearly independent
and give a reason for each answer.
3
a. 2 9
, 1 6
4 12345
b. Columns of 67890
98765
43218 10 3
c. 2
1 9
, 6
3 0
, 0
0 8
d. 2
1
4 11 Characterization of Linearly Dependent Sets
EXAMPLE
Consider the set of vectors v 1 , v 2 , v 3 , v 4 in R 3 in
the following diagram. Is the set linearly dependent? Explain v3
x3 v2
x2 x1 v1
v4 12 Theorem 7
An indexed set S v 1 , v 2 , , v p of two or more vectors is
linearly dependent if and only if at least one of the vectors in
S is a linear combination of the others. In fact, if S is linearly
dependent, and v 1 0, then some vector v j (j 2) is a
linear combination of the preceding vectors v 1 , , v j 1 . 13...
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 Spring '08
 KOSTROV
 Linear Algebra, Algebra, Linear Independence, Vector Space, linear dependence relation

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