2. 6.1 Introduction to Linear Transformations
Function T that maps a vector space V into a vector space W:
space
vector
:
,
,
: mapping
W
V
W
V
T
V: the domain of T
W: the codomain of T
1/36
3. Image of v under T:
If v is in V and w is in W such that
w
v
)
(
T
Then w is called the image of v under T .
the range of T:
The set of all images of vectors in V.
the preimage of w:
The set of all v in V such that T(v)=w.
2/36
4. Ex 1: (A function from R2 into R2 )
2
2
: R
R
T
)
2
,
(
)
,
( 2
1
2
1
2
1 v
v
v
v
v
v
T
2
2
1 )
,
( R
v
v
v
(a) Find the image of v=(-1,2). (b) Find the preimage of w=(-1,11)
Sol:
)
3
,
3
(
))
2
(
2
1
,
2
1
(
)
2
,
1
(
)
(
)
2
,
1
(
)
(
T
T
a
v
v
)
11
,
1
(
)
(
)
(
w
v
T
b
)
11
,
1
(
)
2
,
(
)
,
( 2
1
2
1
2
1
v
v
v
v
v
v
T
11
2
1
2
1
2
1
v
v
v
v
4
,
3 2
1
v
v Thus {(3, 4)} is the preimage of w=(-1, 11).
3/36
5. Linear Transformation (L.T.):
n
nsformatio
linear tra
to
：
:
space
vector
：
,
W
V
W
V
T
W
V
V
T
T
T
v
u
v
u
v
u ,
),
(
)
(
)
(
(1)
R
c
cT
c
T
),
(
)
(
)
2
( u
u
4/36
6. Notes:
(1) A linear transformation is said to be operation preserving.
)
(
)
(
)
( v
u
v
u T
T
T
Addition
in V
Addition
in W
)
(
)
( u
u cT
c
T
Scalar
multiplication
in V
Scalar
multiplication
in W
(2) A linear transformation from a vector space into
itself is called a linear operator.
V
V
T
:
5/36
7. Ex 2: (Verifying a linear transformation T from R2 into R2)
Pf:
)
2
,
(
)
,
( 2
1
2
1
2
1 v
v
v
v
v
v
T
number
real
any
:
,
in
vector
:
)
,
(
),
,
( 2
2
1
2
1 c
R
v
v
u
u
v
u
)
,
(
)
,
(
)
,
(
:
addition
(1)Vector
2
2
1
1
2
1
2
1 v
u
v
u
v
v
u
u
v
u
)
(
)
(
)
2
,
(
)
2
,
(
))
2
(
)
2
(
),
(
)
((
))
(
2
)
(
),
(
)
((
)
,
(
)
(
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2
1
1
2
2
1
1
2
2
1
1
v
u
v
u
T
T
v
v
v
v
u
u
u
u
v
v
u
u
v
v
u
u
v
u
v
u
v
u
v
u
v
u
v
u
T
T
6/36
9. Ex 3: (Functions that are not linear transformations)
x
x
f
a sin
)
(
)
(
2
)
(
)
( x
x
f
b
1
)
(
)
(
x
x
f
c
)
sin(
)
sin(
)
sin( 2
1
2
1 x
x
x
x
)
sin(
)
sin(
)
sin( 3
2
3
2
2
2
2
1
2
2
1 )
( x
x
x
x
2
2
2
2
1
)
2
1
(
1
)
( 2
1
2
1
x
x
x
x
f
2
)
1
(
)
1
(
)
(
)
( 2
1
2
1
2
1
x
x
x
x
x
f
x
f
)
(
)
(
)
( 2
1
2
1 x
f
x
f
x
x
f
( ) sin is not a linear transformation
f x x
2
( ) is not a linear transformation
f x x
( ) 1 is not a linear transformation
f x x
8/36
10. Notes: Two uses of the term “linear”.
(1) is called a linear function because its graph
is a line.
1
)
(
x
x
f
(2) is not a linear transformation from a vector
space R into R because it preserves neither vector
addition nor scalar multiplication.
1
)
(
x
x
f
9/36
11. Zero transformation:
V
W
V
T
v
u,
,
:
V
T
v
v ,
0
)
(
Identity transformation:
V
V
T
: V
T
v
v
v ,
)
(
Thm 6.1: (Properties of linear transformations)
W
V
T
:
0
0
)
(
(1) T
)
(
)
(
(2) v
v T
T
)
(
)
(
)
(
(3) v
u
v
u T
T
T
)
(
)
(
)
(
)
(
)
(
Then
If
(4)
2
2
1
1
2
2
1
1
2
2
1
1
n
n
n
n
n
n
v
T
c
v
T
c
v
T
c
v
c
v
c
v
c
T
T
v
c
v
c
v
c
v
v
10/36
12. Ex 4: (Linear transformations and bases)
Let be a linear transformation such that
3
3
: R
R
T
)
4
,
1
,
2
(
)
0
,
0
,
1
(
T
)
2
,
5
,
1
(
)
0
,
1
,
0
(
T
)
1
,
3
,
0
(
)
1
,
0
,
0
(
T
Sol:
)
1
,
0
,
0
(
2
)
0
,
1
,
0
(
3
)
0
,
0
,
1
(
2
)
2
,
3
,
2
(
)
0
,
7
,
7
(
)
1
,
3
,
0
(
2
)
2
,
5
,
1
(
3
)
4
,
1
,
2
(
2
)
1
,
0
,
0
(
2
)
0
,
1
,
0
(
3
)
0
,
0
,
1
(
2
)
2
,
3
,
2
(
T
T
T
T
T (T is a L.T.)
Find T(2, 3, -2).
11/36
13. Ex 5: (A linear transformation defined by a matrix)
The function is defined as
3
2
: R
R
T
2
1
2
1
1
2
0
3
)
(
v
v
A
T v
v
3
2
into
form
n
nsformatio
linear tra
a
is
that
Show
(b)
)
1
,
2
(
where
,
)
(
Find
(a)
R
R
T
T
v
v
Sol: )
1
,
2
(
)
(
v
a
0
3
6
1
2
2
1
1
2
0
3
)
( v
v A
T
)
0
,
3
,
6
(
)
1
,
2
(
T
vector
2
R vector
3
R
)
(
)
(
)
(
)
(
)
( v
u
v
u
v
u
v
u T
T
A
A
A
T
b
)
(
)
(
)
(
)
( u
u
u
u cT
A
c
c
A
c
T
(vector addition)
(scalar multiplication)
12/36
14. Thm 6.2: (The linear transformation given by a matrix)
Let A be an mn matrix. The function T defined by
v
v A
T
)
(
is a linear transformation from Rn into Rm.
Note:
n
mn
m
m
n
n
n
n
n
mn
m
m
n
n
v
a
v
a
v
a
v
a
v
a
v
a
v
a
v
a
v
a
v
v
v
a
a
a
a
a
a
a
a
a
A
2
2
1
1
2
2
22
1
21
1
2
12
1
11
2
1
2
1
2
22
21
1
12
11
v
v
v A
T
)
(
m
n
R
R
T
:
vector
n
R vector
m
R
13/36
15. Show that the L.T. given by the matrix
has the property that it rotates every vector in R2
counterclockwise about the origin through the angle .
Ex 7: (Rotation in the plane)
2
2
: R
R
T
cos
sin
sin
cos
A
Sol:
)
sin
,
cos
(
)
,
(
r
r
y
x
v
(polar coordinates)
r： the length of v
：the angle from the positive
x-axis counterclockwise to
the vector v
14/36
17. is called a projection in R3.
Ex 8: (A projection in R3)
The linear transformation is given by
3
3
: R
R
T
0
0
0
0
1
0
0
0
1
A
16/36
18. Show that T is a linear transformation.
Ex 9: (A linear transformation from Mmn into Mn m )
)
:
(
)
( m
n
n
m
T
M
M
T
A
A
T
Sol:
n
m
M
B
A
,
)
(
)
(
)
(
)
( B
T
A
T
B
A
B
A
B
A
T T
T
T
)
(
)
(
)
( A
cT
cA
cA
cA
T T
T
Therefore, T is a linear transformation from Mmn into Mn m.
17/36
19. Keywords in Section 6.1:
function: دالة
domain: مجال
codomain: مقابل مجال
image of v under T: التحويل تحت صورة
range of T: T مدى
preimage of w:
linear transformation: خطي تحويل
linear operator: خطي مؤثر
zero transformation: صفري تحويل
identity transformation: محايد تحويل
v
T
18/36
20. 6.2 Matrices for Linear Transformations
)
4
3
,
2
3
,
2
(
)
,
,
(
)
1
( 3
2
3
2
1
3
2
1
3
2
1 x
x
x
x
x
x
x
x
x
x
x
T
Three reasons for matrix representation of a linear transformation:
3
2
1
4
3
0
2
3
1
1
1
2
)
(
)
2
(
x
x
x
A
T x
x
It is simpler to write.
It is simpler to read.
It is more easily adapted for computer use.
Two representations of the linear transformation T:R3→R3 :
19/36
21. Thm 6.10: (Standard matrix for a linear transformation)
such that
on
ansformati
linear trt
a
be
:
Let m
n
R
R
T
,
)
(
,
,
)
(
,
)
( 2
1
2
22
12
2
1
21
11
1
mn
n
n
n
m
m a
a
a
e
T
a
a
a
e
T
a
a
a
e
T
)
(
to
correspond
columns
se
matrix who
Then the i
e
T
n
n
m
.
for
matrix
standard
the
called
is
A
.
in
every
for
)
(
such that
is
T
R
A
T n
v
v
v
mn
m
m
n
n
n
a
a
a
a
a
a
a
a
a
e
T
e
T
e
T
A
2
1
2
22
21
1
12
11
2
1 )
(
)
(
)
(
20/36
24. Ex 1: (Finding the standard matrix of a linear transformation)
by
define
:
L.T.
for the
matrix
standard
the
Find 2
3
R
R
T
)
2
,
2
(
)
,
,
( y
x
y
x
z
y
x
T
Sol:
)
2
,
1
(
)
0
,
0
,
1
(
)
( 1
T
e
T
)
1
,
2
(
)
0
,
1
,
0
(
)
( 2
T
e
T
)
0
,
0
(
)
1
,
0
,
0
(
)
( 3
T
e
T
2
1
)
0
0
1
(
)
( 1
T
e
T
1
2
)
0
1
0
(
)
( 2
T
e
T
0
0
)
1
0
0
(
)
( 3
T
e
T
Vector Notation Matrix Notation
23/36
25.
0
1
2
0
2
1
)
(
)
(
)
( 3
2
1 e
T
e
T
e
T
A
Note:
z
y
x
z
y
x
A
0
1
2
0
2
1
0
1
2
0
2
1
y
x
y
x
z
y
x
z
y
x
A
2
2
0
1
2
0
2
1
)
2
,
2
(
)
,
,
(
i.e. y
x
y
x
z
y
x
T
Check:
24/36
26. Ex 2: (Finding the standard matrix of a linear transformation)
.
for
matrix
standard
the
Find
axis.
-
x
the
onto
in
point
each
projecting
by
given
is
:
n
nsformatio
linear tra
The
2
2
2
T
R
R
R
T
Sol:
)
0
,
(
)
,
( x
y
x
T
0
0
0
1
)
1
,
0
(
)
0
,
1
(
)
(
)
( 2
1 T
T
e
T
e
T
A
Notes:
(1) The standard matrix for the zero transformation from Rn into Rm
is the mn zero matrix.
(2) The standard matrix for the zero transformation from Rn into Rn
is the nn identity matrix In
25/36
27. Ex 3: (The standard matrix of a composition)
s.t.
into
from
L.T.
be
and
Let 3
3
2
1 R
R
T
T
)
,
0
,
2
(
)
,
,
(
1 z
x
y
x
z
y
x
T
)
,
z
,
(
)
,
,
(
2 y
y
x
z
y
x
T
,
'
and
ns
compositio
for the
matrices
standard
the
Find
2
1
1
2 T
T
T
T
T
T
Sol:
)
for
matrix
standard
(
1
0
1
0
0
0
0
1
2
1
1 T
A
)
for
matrix
standard
(
0
1
0
1
0
0
0
1
1
2
2 T
A
26/36
28. 1
2
for
matrix
standard
The T
T
T
2
1
'
for
matrix
standard
The T
T
T
0
0
0
1
0
1
0
1
2
1
0
1
0
0
0
0
1
2
0
1
0
1
0
0
0
1
1
1
2 A
A
A
0
0
1
0
0
0
1
2
2
0
1
0
1
0
0
0
1
1
1
0
1
0
0
0
0
1
2
' 2
1A
A
A
27/36
29. Inverse linear transformation:
in
every
for
s.t.
L.T.
are
:
and
:
If 2
1
n
n
n
n
n
R
R
R
T
R
R
T v
))
(
(
and
))
(
( 2
1
1
2 v
v
v
v
T
T
T
T
invertible
be
to
said
is
and
of
inverse
the
called
is
Then 1
1
2 T
T
T
Note:
If the transformation T is invertible, then the inverse is
unique and denoted by T–1 .
28/36
30. Ex 5: (Finding a matrix relative to nonstandard bases)
by
defined
L.T.
a
be
：
Let 2
2
R
R
T
)
2
,
(
)
,
( 2
1
2
1
2
1 x
x
x
x
x
x
T
)}
1
,
0
(
),
0
,
1
{(
'
and
)}
1
,
1
(
),
2
,
1
{(
basis
the
to
relative
of
matrix
the
Find
B
B
T
Sol:
)
1
,
0
(
3
)
0
,
1
(
0
)
3
,
0
(
)
1
,
1
(
)
1
,
0
(
0
)
0
,
1
(
3
)
0
,
3
(
)
2
,
1
(
T
T
3
0
)
1
,
1
(
,
0
3
)
2
,
1
( '
' B
B T
T
'
and
to
relative
for
matrix
the B
B
T
3
0
0
3
)
2
,
1
(
)
2
,
1
( '
' B
B T
T
A
29/36
31. Ex 6:
)
1
,
2
(
where
),
(
find
to
matrix
the
use
5,
Example
in
given
：
L.T.
For the 2
2
v
v
T
A
R
R
T
Sol:
)
1
,
1
(
1
)
2
,
1
(
1
)
1
,
2
(
v
1
1
B
v
3
3
1
1
3
0
0
3
)
( ' B
B A
T v
v
)
3
,
3
(
)
1
,
0
(
3
)
0
,
1
(
3
)
(
v
T )}
1
,
0
(
),
0
,
1
{(
'
B
)}
1
,
1
(
),
2
,
1
{(
B
)
3
,
3
(
)
1
2(2)
,
1
2
(
)
1
,
2
(
T
Check:
30/36
32. Keywords in Section 6.2:
standard matrix for T: الخطي للتحويل األساسية المصفوفة
composition of linear transformations: الخطية للتحويالت تركيب
inverse linear transformation: عكسي خطي تحويل
matrix of T relative to the bases B and B‘ :
الخطي التحويل مصفوفة
T
لألساس بالنسبة
B
و
B’
matrix of T relative to the basis B:
الخطي التحويل مصفوفة
T
لألساس بالنسبة
B
T
31/36
33. 6.3 Similarity
Ex 4: (Similar matrices)
similar
are
2
1
2
3
'
and
3
1
2
2
)
(
A
A
a
1
0
1
1
where
,
'
because 1
P
AP
P
A
similar
are
3
1
1
2
'
and
7
3
7
2
)
(
A
A
b
1
2
2
3
where
,
'
because 1
P
AP
P
A
32/36
34. Ex 5: (A comparison of two matrices for a linear transformation)
basis.
standard
the
to
relative
:
for
matrix
the
is
2
0
0
0
1
3
0
3
1
Suppose 3
3
R
R
T
A
)}
1
,
0
,
0
(
),
0
,
1
,
1
(
),
0
,
1
,
1
{(
'
basis
the
to
relative
for
matrix
the
Find
B
T
Sol:
1
0
0
0
1
1
0
1
1
)
1
,
0
,
0
(
)
0
,
1
,
1
(
)
0
,
1
,
1
(
matrix
standard
the
to
from
matrix
n
transitio
The
B
B
B
P
B'
1
0
0
0
0
2
1
2
1
2
1
2
1
1
P
33/36
36. Notes: Computational advantages of diagonal matrices:
n
d
d
d
D
0
0
0
0
0
0
2
1
k
n
k
k
k
d
d
d
D
0
0
0
0
0
0
)
1
( 2
1
D
DT
)
2
(
0
,
0
0
0
0
0
0
)
3
(
1
1
1
1 2
1
i
d
d
d
d
D
n
35/36
37. Keywords in Section 6.3:
matrix of T relative to B:
الخطي التحويل مصفوفة
T
لألساس بالنسبة
B
matrix of T relative to B' :
الخطي التحويل مصفوفة
T
لألساس بالنسبة
B’
transition matrix from B' to B :
األساس من االنتقال مصفوفة
B’
الى
B
transition matrix from B to B' :
األساس من االنتقال مصفوفة
B
الى
B’
similar matrix:
مماثلة مصفوفة
36/36
