Wednesday February 11
−
Lecture 17 :
Examples on
linear independence
(Refers to
section 4.2)
Expectations
:
1.
Define a linearly independent set.
2.
Determine when a subset of
R
n
is linearly independent.
17.1
Theorem
−
Let
S
= {
v
1
,
v
2
, …,
v
k
} be a set of
k
distinct nonzero vectors in
R
n
.
If
k
>
n
then the set
S
cannot be linearly independent.
Proof:
•
Let
A
be the matrix whose columns are
v
1
,
v
2
, …,
v
k
.
•
Then
A
is an
n
by
k
matrix.
•
If
k
>
n
then
A
has more columns then rows and so
A
RREF
must point to at least
one free variable in the system
A
x
=
0
.
•
Hence
A
x
=
0
has infinitely many solutions and so has nontrivial solutions.
Thus any set with more vectors then the dimension of
R
n
is not linearly independent.
•
Thus if
S
= {
v
1
,
v
2
, …,
v
k
} is linearly independent in
R
n
then
k
≤
n
.
17.2
Example
−
Show that the set {
e
x
,
x
2
} is linearly independent in the vector space of
all realvalued continuous functions on
R
.
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 Winter '08
 All
 Linear Algebra, Derivative, Linear Independence, Vector Space, linearly independent set, realvalued continuous functions

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