Section 1.8 : An Introduction to Linear
Transforms
Chapter 1 : Linear Equations
MATH - UH 1022 Linear Algebra
Section 1.8
Slide 1
1.8 : An Introduction to Linear Transforms
Topics
We will cover these topics in this section.
1. The definition of a linear transformation.
2. The interpretation of matrix multiplication as a linear
transformation.
Objectives
For the topics covered in this section, students are expected to be able to
do the following.
1. Construct and interpret linear transformations in Rn (for example,
interpret a linear transform as a projection, or as a shear).
2. Characterize linear transforms using the concepts of
I existence and uniqueness
I domain, co-domain and range
Section 1.8
Slide 2
From Matrices to Functions
Let A be an m ⇥ n matrix. We define a function
I
can
also
T : Rn ! Rm ,
T (~v ) = A~v
mxn
-> A8
4
This is called a matrix transformation.
be
denoted
by TE
• The domain of T is Rn .
• The co-domain or target of T is Rm .
• The vector T (~x) is the image of ~x under T
• The set of all possible images T (~x) is the range.
This gives us another interpretation of A~x = ~b:
• set of equations
• augmented matrix
• matrix equation
• vector equation
• linear transformation equation
Section 1.8
Slide 3
-
b RM
belong
T'
to rage of
= T()
=
Fa
↳Ya
What dos it
for
mean
->
t."
of
b
raye
T
:
"
*
-
->
RM
A*
->
Range (t)
B
B
=
=
:
T(x)
A
ER"
3
such
:
-
is
consistent !
(A
[Si]
=
Tp
IR"
"
:
1R
-
(32)(5) ( 3)
=
x e)
-
,
(x
=
in 10 10)
in
(8)
[si)() (8]
=
=
=
8
+
2y , 3x
43)
+
10 07
=
,
.
Note that this is
always
true
(linear transformations always
:
A 8
read
=
:
to
0
-
T
R
:
R
+
(x
(x y)
,
y, y
+
i,=CoI a
+
1)
(0
(2)
A
.
TA
:
R
2
-
# of
How?
transformin
of
of
[ == -]
=
TA(X
linear
t
,
3,
z)
R
->
#
of
A
columns
(x
+
[ii -][E]
=
-
z
,
rows
2x
3y
+
(2x3y z]
+
+
A
.
z)
.
(3)
T
2
:
R
(x , y)
Find
Asxz
-
RRY
+
(x
+
2y , y
such that I
C= 73]= <I
?
=
n
-
TA
:
,
y
n)
+
Functions from Calculus
Many of the functions we know have domain and codomain R.We can
express the rule that defines the function sin this way:
f: R!R
f (x) = sin(x)
In calculus we often think of a function in terms of its graph, whose
horizontal axis is the domain, and the vertical axis is the codomain.
y
1
⇡
sin(x)
0
⇡
2⇡
x
This is ok when the domain and codomain are R. It’s hard to do when
the domain is R2 and the codomain is R3 . We would need five
dimensions to draw that graph.
Section 1.8
Slide 4
Example 1
2
1
Let A = 40
1
3
2 3

1
7
3
15, ~u =
, ~b = 455.
4
1
7
a) Compute T (~u).
b) Calculate ~v 2 R2 so that T (~v ) = ~b
c) Give a ~c 2 R3 so there is no ~v with T (~v ) = ~c
or: Give a ~c that is not in the range of T .
or: Give a ~c that is not in the span of the columns of A.
Section 1.8
Slide 5
A
=
=
(a)
(b)
.
:
(i , ]
(i)
=
(i)
Look
E
for
such that
T
i
=
=
=
I
[4]
.
=
,
(d
=
2
Observe
-
?)
(
such that
is
]
(
( !)(i)
I
choose I
1R
=
I
no
T(E)
Let
(]
=
Such
is
not
that
+
B
B
+
3
2
=
B
c
in the
=
=
,
3
+(3
:
range of
J
.
EX
A
:
C !: ]
=
AX
x
T
:
IR3
Describe the
T
.
I
effect of
i
-
Let
i(*)
->
=
R3
A
=
the
[]
*
)
:
(
:
/]
[
=
I
E
transformation
Linear Transformations
A function T : Rn ! Rm is linear if
Linear
• T (~u + ~v ) = T (~u) + T (~v ) for all ~u, ~v in R .
n
• T (c~v ) = cT (~v ) for all ~v 2 R , and c in R.
n
So if T is linear, then
transformation
preserve
the
of rector addification
T (c1~v1 + · · · + ck~vk ) = c1 T (~v1 ) + · · · + ck T (~vk )
This is called the principle of superposition. The idea is that if we
know T (~e1 ), . . . , T (~en ), then we know every T (~v ).
Fact: Every matrix transformation TA is linear.
↳>
Section 1.8
Slide 6
operations
Proof
:
*
R
Ta
Proof
a)
:
Let
:
->
In
Tax)
+
Ta(n F)
Amxn
A
:
transf
⑭"
TA(n 8)
,
Let
linear
is
i
W TS
.
Ni
-
Ta(n)
=
A(a ) .
=
+
TACE)
.
+
+
=
=
sec
A+ Ar
Ta(n)
+
Ta(E)
.
1 3
.
.
b)
let
= RY
I(v)
↳
Wit
.
S
TA((r)
=
=
=
=
A(c)
G
cA(E)
Tz(v)
.
Let T
be
a
T
matrix
transformation
*
:
T(E)
M
=
->
A
Rm
*
Im
⑲in
-
Transformation
⑧
Linear
Tit
Matrix
functions
IR*
->
IRM
Observation
I
T
is
:
transformation
linear
a
,
then T(8) =
Proof
.
T(8)
i(0a)
=
=
=
or
0T(i)
0
+(8) T(w ( =)
+
-
=
T()
=
=
-
T()
8
.
+
T( n)
-
-
T(i)
Is
the
true ?
converse
T(0)
#
=
that T is
8
,
does it
linear
a
imply
transformation
Sin (x+y) sinxcosy + coxsi
=
R
T
x
:
i(x
+
R
-
x
-
y)
(x y)
*
+
=
=
?
-
T
:
x ->
TM
=
sin
sin(x)
(x)
x"
+
T(x)
T(x+y)
+
yT1y)
= TK)+Tle)
Observation :
the
transformation
If
T(x BE) xT(n) BT(t)
T is
a
linear
+
=
*
+
for
x
,
all
in
i
and
all scalars
what
about the
2
,
The
,
,
of
95
.
?
*
<pp)
(pT(ni)
T(ni) xT(ii)
T(x n?
d
B
converse
Repeated application
=
domain
the
+
2
+
+
...+
+
-
-
-
euperposition principle !
+
Why
is
interesting
it
suppose
-
i
T(a)
Conclusion
is known
it
is
=
,
Span di
x,
+
linear
a
x
=
deal
transformations
linear
Let T be
to
T(wi)
If
:
a
Oh
known
on
Span GE?
+
-
+
-
,
?
=2 ,53
xgs
:
-
transformation
x2
+(v)
linear
,
E
transf
s
rector
,
xsT(v)
+
-ormation
,
any
,
with
]
-
Then
,
in
Example 2
Suppose T is the linear transformation T (~x) = A~x. Give a short
geometric interpretation of what T (~x) does to vectors in R2 .

0 1
1) A =
1 0
-
2) A =

1 0
0 0

k
0
-
0
3) A =
Section 1.8
Slide 7
0
for k 2 R
k
Example 3
What does TA do
2
1 0
a) A = 40 1
0 0
2
1
b) A = 40
0
Section 1.8
Slide 8
to vectors in R3 ?
3
0
05
0
3
0 0
1 05
0 1
Example 4
A linear transformation T : R2 7! R3 satisfies
2 3
2 3
✓ ◆
✓ ◆
5
3
1
0
T
= 4 75 ,
T
= 4 85
0
1
2
0
What is the matrix that represents T ?
·
·
Section 1.8
Describe the
Is T
Slide 9
=
TA
·
transformation
A
=
?
T
.
We look
T
=
for A
such that
3x2
TA
-