NOTES ON SECTION
4
.
4

4
.
6
MA265, SECTIONS 41, 52
Some key words
basis
dimension
linearly dependent
linearly independent
span
spanning set
1.
Overview
We want two things in this set of section:
•
A measure for how large a vector space is (ultimately provided by dimension).
•
A system to parameterize elements in a vector space by coordinates (provided by a basis).
They are related: bases are used to calculate dimension. But you should also understand that bases do much
more than that: they provide a coordinate system that allows one to parameterize the vector space by real
numbers, thus allowing us to compare the vector space to
R
n
. We don’t pursue this as far as possible, but
I strongly encourage you to at least read section 4
.
8 to get some of the details of what I mean by this in a
more precise fashion.
As we will see, dimension is a count of the number of parameters needed to describe a vector space
(alternative, the number of independent directions it has). There are two questions to ask
(1) Given a collection of parameters (directions) describing elements in the vector space, we ask: “Do I
have enough parameters (directions)?”
(2) We also ask: “Do I have too many parameters (redundant or dependent directions)?”
The first question is clarified by the notion of “span”, the second by the notion of ”linear in/depenendence”.
2.
Span
Definition 1.
We defined span already in
4
.
3
.
Definition 2.
A set of vectors
spans
V
if span
(
S
) =
V
. In this case
S
is called a
spanning set
for
V
.
Example 3.
Take
V
=
R
3
. The set of vectors
S
=
1
0
0
,
0
1
0
,
0
0
1
spans
R
3
. You can make the obvious generalization to get spanning sets of
R
n
in general.
Example 4.
Take
V
=
M
2
×
2
, the set of
2
×
2
matrices with real coefficients (itself a vector space). It has
the following spanning set
S
=
1
0
0
0
,
0
1
0
0
,
0
0
1
0
,
0
0
0
1
Example 5.
If
S
is a spanning set for a vector space
V
, so is any set
S
0
which contains
S
: the reason is
that if you can build any vector out of the vectors in
S
, and everything in
S
is in
S
0
, then surely you can
build any vector out of the vectors in
S
0
(just forget about the extra vectors you added to get from
S
to
S
0
.
For instance, in Example 3 the set
S
0
S
0
=
1
0
0
,
0
1
0
,
0
0
1
,
1
1
0
,
0
1

1
,
1
2
1
also spans.
1
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In this way, span allows us to determine if we have enough vectors: enough means that we can build any
vector out of the vectors in
S
using linear combinations. But it doesn’t tell us if we have too many (which
is why we’ll need linear dependence later).
It is useful to have a criteria in
R
n
to determine if a set of vectors spans
R
n
: here is the answer
Theorem 6.
If
S
is a set of vectors, and
A
is the matrix obtained by putting the vectors in
S
as the columns
of a matrix, then
S
spans
R
n
if and only if the RREF of
A
has no row of zeros.
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 Spring '08
 Bens
 Linear Algebra, Algebra, Vectors, Vector Space

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