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Friday January 16
−
Lecture 7
:
Standard matrix of a transformation
.
(Refers to
section 3.3
)
Expectations
:
1.
Find the standard matrix induced by a linear transformation.
Properties of linear transformations
.
7.1
Definition
−
Let {
e
1
,
e
2
, .
..,
e
n
} be a set of vectors in
R
n
where
e
i
is the vector
containing only zeros except for the
i
th
entry, which is a 1.
•
We call this set of vectors the
standard basis
of
R
n
.
7.2
Theorem
−
Let
T
be a function mapping vectors from
R
n
into
R
m
. If
T
is a linear
transformation then there exists a unique
n
x
m
matrix
A
such that
T
(
x
) =
A
x
for all
x
in
R
n
. In this case,
A
= [
c
1
c
2
...
c
n
]
where each column vector
c
i
=
T
(
e
i
) ,
i
= 1 to
n
.
Proof:
•
Suppose
T
is a linear transformation. We are required to show that there exists a
matrix
A
such
T
(
x
) =
A
x
for all
x
in
R
n
.
•
Suppose
x
= (
x
1
,
x
2
,
x
3
, .
..,
x
n
) in
R
n
. Then
x
=
x
1
e
1
+
x
2
e
2
+ .
.. +
x
n
e
n
.
•
For each
i
let
c
i
=
T
(
e
i
).
•
Then
T
(
x
) =
T
(
x
1
e
1
+
x
2
e
2
+ .
.. +
x
n
e
n
) =
x
1
T
(
e
1
) +
x
2
T
(
e
1
) + .
.. +
x
n
T
(
e
n
). =
x
1
c
1
+
x
2
c
2
+ .
.. +
x
n
c
n
.
•
If
A
is the matrix [
c
1
c
2
...
c
n
] whose columns are the vectors
c
i
, then
T
(
x
) =
A
x
.
•
Hence the matrix
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This note was uploaded on 09/04/2009 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.
 Winter '08
 All
 Linear Algebra, Algebra, Transformations

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