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Friday January 16
−
Lecture 6
:
Matrices as linear transformations
.
(Refers to
section 3.2)
Expectations:
1.
Define a transformation
T
from
R
n
to
R
m
.
2.
Define the
kernel
and
range
of a linear transformation.
3.
Given a matrix
A
, show that it can be viewed as a linear transformation
T
(
x
) =
A
x
.
4.
Define the standard matrix for a transformation
T
:
R
n
→
R
m
.
5.
If
T
(
x
) =
A
x
,
A
standard matrix of
T
, then
T
maps
R
n
onto R
m
6.1
Definition
−
A
transformation
is a function mapping
R
n
to
R
m
. A
linear
transformation
T
from
R
n
to
R
m
is a rule which assigns to each vector
a
in
R
n
a unique
vector denoted by
T
(
a
)
in
R
m
satisfying the condition:
T
(
α
u
+
β
v
) =
α
T
(
u
) +
β
T
(
v
)
for any pair of vectors
u
and
v
in
R
n
and scalars
α
and
β
.
We sometimes say that
T
is
linear if it “
respects linear combinations
”.
6.1.1
Definitions
−
Given a linear transformation
T
:
R
n
→
R
m
,
•
R
n
is called the
domain
of
T
, and since
T
maps vectors into
R
m
we call
R
m
the
codomain
.
•
T
[
R
n
] = {
T
(
u
) :
u
∈
R
n
} is called the
range
of
T
or
image
of
R
n
under
T
.
The vector
T
(
u
) is the
image
of the vector
u
under
T
.
6.1.2
Example
−
Verify whether the map
f
(
x
) = |
x
| is linear. What about the map
g(
x
1
,
x
2
) = (
x
1
2
,
x
1
x
2
)?
6.1.3
Definition
−
If
T
:
R
n
→
R
m
is a transformation the set {
u
∈
R
n
:
T
(
u
) =
0
} is
called the
kernel
or the
null space
of
T
.
6.2
The matrix
A
m
×
n
viewed as a linear transformation from R
n
into
R
m
.
Recall that from the properties of matrix algebra, given an matrix
A
m
×
n
, any two vectors
u
and
v
in
R
n
and any two scalars
α
and
β
we have

This
** preview**
has intentionally

A
(
α
u
+
β
v
) =
A
(
α
u
) +
A
(
β
v
) =
α
A
u
+
β
A
v
a vector in
R
m
.
•
Hence the matrix
A
m
×
n
can be viewed as a function sending a vector
u
in
R
n
to a
vector
A
u
in
R
m
. So the matrix
A
m
×
n
can be viewed as function which
"
transforms
" vectors in
R
n
to vectors in
R
m
. The matrix
A
m
×
n
:
R
n
→
R
m
is a
transformation
A
:
•
Not only that, it does so "linearly".
•
It can be viewed as a linear transformation from
R
n
into
R
m
. When viewed in
this way we call it a
matrix transformation
.
6.3
Examples
1) Consider the matrix
A
⎡
1
0
⎤
⎣
0
−
1
⎦
Viewed as a linear transformation it will map vectors in
R
2
to vectors in
R
2
.
•
We investigate to what vector it will map a vector (
x
,
y
).
We see that
⎡
1
0
⎤
⎡
x
⎤
⎡
x
⎤
⎣
0
−
1
⎦
⎣
y
⎦
=
⎣
−
y
⎦
•
We see that this particular matrix "reflects" vectors in
R
2
about the
x
-axis.
•
This is why we call this particular matrix the "

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