SECTION 4.3 BASES FOR VECTOR SPACES
Now, if we have a vector space
V
, we
CANNOT
talk about systems of equations, column
vectors, matrices, or row operations. However, we
CAN
talk about the following for lists of
vectors
v
1
, . . . ,
v
p
in
any
vector space
V
.
SPAN:
The span of the list is Span
{
v
1
, . . . ,
v
p
}
, which is the set of all linear combinations
of
v
1
, . . . ,
v
p
.
LINEAR INDEPENDENCE:
The ONLY weights for which
c
1
v
1
+
c
2
v
2
+
· · ·
+
c
p
v
p
=
0
are
c
1
=
c
2
=
· · ·
=
c
p
= 0.
LINEAR DEPENDENCE:
There are weights,
not all zero
, for which
c
1
v
1
+
c
2
v
2
+
· · ·
+
c
p
v
p
=
0
. This equation is then called a
linear dependence relation.
BASIS:
A
basis
for a subspace
H
of a vector space
V
is an ordered list of vectors
B
=
{
b
1
, . . . ,
b
p
}
which are linearly independent and span
H
. Remember that
H
might
be
V
itself.
EXAMPLES.
(1) The standard basis for
R
5
(2) The standard basis for
P
7
, the vector space of polynomials of degree no larger than 7.
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(3) The columns of an invertible
n
×
n
matrix
A
are a basis for
R
n
. Why?
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 Spring '08
 PAVLOVIC
 Linear Algebra, Vector Space

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