Math
115

Week
2
Findabasisforsubspaces
Lecture
47
Example
:
Determine
if
p
{
x
2x
'
,
x
x
'
,
2
x
'
}
is
linearly
independent
TY
Fix
,
Tax
)
We
need
to
determine
if
we
obtain
the
t
rival
solution
to
c
,
pi
(
x
)
capzlx
)
Csps
(
x
)
,
a
,
a.
GEIR
e
,
X
XZ
Grouping
the
Xo
,
X
,
X
'
terms
,
we
obtain
c
,
(
x
2x
'
)
a
C
x
x
)
↳
(
2
x
'
)
(
c
,
2cg
)
(
c
,
Cz
)
x
(
2C
,
Cz
Cs
)
x
x
'
Putting
this
into
a
matrix
and
row
reducing
gives
2
2
2
Rz
Ri
Rst
R2
2
2
Rs
212
,
2
5
7
each
column
has
the
coefficient
of
each
polynomial
(
a
box
CX
)
becomes
§
Since
the
matrix
is
upper
triangular
,
so
it
can
be
reduced
to
s
giving
the
t
rival
solution
,
i.
e.
,
p
is
linearly
independent
man
We
will
go
through
some
examples
showing
how
to
find
a
basis
for
a
subspace
of
a
vector
space
Example
:
find
a
basis
for
the
subspace
defined
by
a
b
S
{
ca
Cb
ER
'
a
,
b.
CER
}
.
2
Span
{
,
,
}
Span
{
,
}
and
so
C
is
a
basis