1
Span of a set of vectors
Properties:
Span{
u
} = the set of all multiple of
u
, and Span{
0
} = {
0
}.
S
contains a nonzero vector.
⇒
Span
S
has infinitely many vectors.
Example:
:
⇒
Span
S
3
= Span
S
4
=
R
2
nonparallel vectors
Example:
Span{
e
1
,
e
2
} =
xy
plane in
R
3
Span{
e
3
} =
z
axis in
R
3
Example:
∈
Span
S
?
A
= [
"
]
The reduced row echelon
form of [
A
v
] is
The reduced row echelon
form of [
A
w
] is
⇒
v
∈
Span
S
⇒
w
∉
Span
S
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Definition: If
S
,
V
⊂
R
n
, and Span
S
=
V
, then
S
is a
generating set
for
V
, or
S
generates
V
.
Examples:
generates
R
3
?
A
= [
"
]
for any
v
in
R
3
, let [
R
c
] be the reduced row echelon form of
[
A
v
], then
⇒
[
R
c
] has no nonzero rows with only entries from
c
⇒
for any
v
in
R
3
,
v
=
A
x
for some
x
, thus Span
S
=
R
3
.
: rank = 3
(e)
There is a pivot position in each row of
A
.
Proof
(a)
⇔
(b): for any
b
in
R
m
,
b
=
A
x
for some
x
.
(c)
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 Spring '09
 Fong
 ax, Row echelon form, Spans

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