Monday February 9
−
Lecture 16:
Linear independence
(Refers to section 4.2)
Expectations
:
1.
Define a linearly independent set.
2.
Recognize that subsets of linearly independent sets are linearly independent.
3.
Characterize a linearly independent set as one being a set where no vector is a
linear combination of the others.
4.
Recognize that a matrix
A
x
=
0
only has the trivial solution iff its columns are
linearly independent.
5.
Determine when a subset of
R
n
is linearly independent.
16.1
Definition – Let {
v
1
,
v
2,
....
,
v
k
} be a spanning family for a vector space
V
. Then for
every vector
v
in
V
there exists scalars
α
1
,
α
2
,
.....
,
α
k
such that
v
=
α
1
v
1
+
α
2
v
2
+
.....
+
α
k
v
k
.
If for every vectors
v
this set of scalars
α
1
,
α
2
,
...,
α
k
associated to
v
in this way is
unique
then we say that this spanning family satisfies the
unique representation property
.
16.1.1
Example – The spanning family {(1, 0), (0,1)} = {
e
1
,
e
2
} satisfies the unique
representation property since for an arbitrary vector
v
= (
a
,
b
) in
R
2
,
a
and
b
are the
unique scalars associated to
v
so that
v
=
a
e
1
+
b
e
2
.
16.1.2
Example – The family of vectors
{
v
1
,
v
2
,
v
3
} = {
x
3
+
x
2
+ 1,
2
x
3
+ 1/2,
3
x
3
−
x
2
}
in
P
3
(the family of all polynomials of degree 3 or less including the 0polynomial)
does not satisfy the
unique representation
property
since the vector
0
(the 0
polynomial) has
two
ways of representing it as a linear combination of
v
1
,
v
2
and
v
3
:
0
=
1(
x
3
+
x
2
+ 1)
+ (–2) (2
x
3
+ ½) + 1(3
x
3
−
x
2
)
=
v
1
+ (–2)
v
2
+ 1
v
3
and
0
=
0(
x
3
+
x
2
+ 1)
+ 0(2
x
3
+ ½) + 0(3
x
3
−
x
2
)
= 0
v
1
+ 0
v
2
+ 0
v
3
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So we have two sets of scalars {1, –2, 1} and {0, 0, 0} associated to the
0
vector
in this way. s
16.2
Definition – Let
v
1
,
v
2,
....
,
v
k
be
k
vectors in a vector space
V
.
The vectors
v
1
,
v
2,
....
,
v
k
are said to be
linearly independent
if the ONLY way that
α
1
v
1
+
α
2
v
2
+
.....
+
α
k
v
k
=
0
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 Winter '08
 All
 Linear Algebra, Algebra, Linear Independence, Vector Space, Sets, Linear combination, vk

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