1
Lecture 6
Thm: Let
V
V
,
be finitedimensional vector spaces and
V
V
T
:
is an isomorphism.
If
n
b
b
b
B
...,
,
,
2
1
is any basis of V, then
)
(
...,
),
(
),
(
2
1
n
b
T
b
T
b
T
is a basis
of
V
.
6.1 Matrix representation of transformation
Review Section 2.3
Thm: Let T: R
n
→ R
m
be a linear transformation. Then, there is an unique matrix A
M
m,n
(R)
such that
x
A
x
T
)
(
for
x
R
n
where



)
(
...,
),
(
),
(



2
1
n
e
T
e
T
e
T
A
 the standard matrix representation of T.
and
n
e
e
e
...,
,
,
2
1
is the standard basis for R
n
Example: Let T: R
2
→ R
3
be a linear transformation defined by
2
1
1
2
1
2
1
2
4
3
2
x
x
x
x
x
x
x
T
.
Find the standard matrix representation A of T.
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Thm: Let
V
V
,
be finitedemensional vector spaces and,
n
b
b
b
B
...,
,
,
2
1
and
n
b
b
b
B
...,
,
,
2
1
be an ordered bases for
V
V
,
, respectively. Let
V
V
T
:
be
a linear transformation and
m
n
R
R
T
:
be the linear transformation such that for each
V
v
, we have
B
B
v
T
v
T
)
(
)
(
. Then the standard matrix representation of
T
is the
matrix A whose jth column vector is
B
i
b
T
)
(
, and
B
B
v
A
v
T
)
(
for all vectors
V
v
.
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 Fall '09
 YANG
 Linear Algebra, Algebra, Vector Space, Dot Product, Euclidean space, inner product, Inner product space

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