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Linear Transformation
Linear Transformation: Let V and U be two vector spaces over the same scalar field F. Then
the mapping T^ : V^ U is called a linear transformation or linear mapping if it satisfies the conditions:
1. For any u v , V , T u v T u T v
2. For any a F and v V , T av aT v
Alternatively, A mapping :
n m
T R R defined by T v Mv , where M is an m n real matrix;
is called a linear transformation.
Transformation matrix: If a linear mapping :
n m
T R R defined by T v Mv , where M
is an m n real matrix; then the transformation matrix is T M.
Problem-01: Test whether the mapping
3 2
T : R R defined by T x y z , , x 2 , y y 5 z is linear
or not.
Solution: We have,
3 2 T : R R
And T x y z , , x 2 , y y 5 z
So, we must need a 2 3 matrix consisting of real numbers only.
The given transformation can be written as,
x x y T y
y z z
^
^
x y z
x y z
^
x
y
z
Mv
Since,
T M
is a matrix of order 2 ^3 , so we conclude that T is a linear
transformation.
Problem-02: Test whether the mapping
3 2
T : R R defined by T x y z , , x y 5 , z yz is linear or
not.
Solution: We have,
3 2 T : R R
So, we must need a 2 3 matrix consisting of real numbers only.
The given transformation can be written as,
x
x y z T y yz z
^
^
^
x
y z z
Since the matrix contains variable so M does not exist. We conclude that T is not a linear
transformation.
Problem-03: Check the followings for linear transformations and find
1 T
if it exists. Also verify for
a non-zero vector.
3 3 T : R R is given by T x y z , , (^) (^) x 2 , y 2 , z z x
3 2 T : R R is given by T x y z , , x 2 ,6 y z 5 x
3 2 (^) T : R R is given by T x y z , , (^) (^) 23 x 2 y 5 ,0 z
3 4 (^) T : R R is given by T x y z , , (^) (^) 8 , x z ,4 z 5 ,5 x
3 3 T : R R is given by T x y z , , (^) (^) 2 x y ,4 x y ,3 z
5 3 T : R R is given by T^ p q r s t , , , ,^^ ^ s^ t ,0,5
3 4 (^) T : R R is given by T (^) p q r , , (^) (^) p q ,0, z y ,
3 3 T : R R is given by T^ p q r , ,^^ ^ 3 x^ ^ y ,^^ ^2 x^ ^4 y^ ^ 3 ,5 z^^ x^ ^4 y^ ^2 z
Solution: 1). We have,
3 3 T : R R
And T x y z , , (^) (^) x 2 , y 2 , z z x
So, we must need a 3 3 matrix consisting of real numbers only.
The given transformation can be written as,
x x y
T y (^) z
z z x
x y z
x y z
x y z
^
x
y
z
Mv
And
1 2 1 ^2 ^2 ^2 1, 2, 2 2, , 2 2 4 2 2
T
(^) (^)
1,0,1 ( Verified ).
2). We have,
3 2 T : R R
And T (^) x y z , , (^) x 2 ,6 y z 5 x
So, we must need a 2 ^3 matrix consisting of real numbers only
The given transformation can be written as,
x
x y T y z x z
^ ^
^
^
x y z
x y z
x
y
z
^
Mv
Since,
T M
^
is a matrix of order 2 ^3 , so we conclude that T is a linear
transformation.
Again, since the transformation matrix (^) T is not square, so (^) T is not possible. So that
1 T
does not
exist. ( Ans .)
3). We have,
3 2 T : R R
And T x y z , , (^) (^) 23 x 2 y 5 ,0 z
So, we must need a 2 ^3 matrix consisting of real numbers only.
The given transformation can be written as,
x
x y z T y
z
^ ^
^
^
x y z
x y z
x
y
z
Mv
Since,
T M
is a matrix of order 2 ^3 , so we conclude that T is a linear
transformation.
Again, since the transformation matrix (^) T is not square, so (^) T is not possible. So that
1 T
does not
exist. ( Ans .)
4). We have,
3 4 T : R R
And T x y z , , (^) (^) 8 , x z ,4 z 5 ,5 x
So, we must need a 4 ^3 matrix consisting of real numbers only.
The given transformation can be written as,
x x z T y z x z
^
x y z
x y z
x y z
^
Since, the transformation matrix T^ M does not exist. So, we conclude that T is not a linear
transformation. ( Ans. )
5). We have,
3 3 T : R R
And T x y z ,^ ,^^ ^ 2 x^ ^ y ,4^^ x^ y ,3^ z
So, we must need a 3 3 matrix consisting of real numbers only.
The given transformation can be written as,
x (^) x y
T y x y
z (^) z
^
x y z
x y z
x y z
^
x
y
z
Mv
T (^) 1,0,1 (^) 2.1 0,4.1 0,3.1 2,4,3
And
1 2 4 2.2 4 3 2, 4,3 , , 1,0, 6 3 3
T
(^)
( verified )
6). We have,
5 3 T : R R
And T (^) p q r s t , , , , (^) (^) s t ,0,5
So, we must need a 3 5 matrix consisting of real numbers only.
The given transformation can be written as,
0
5
p
q s t
T r
s
t
^
p q r s t
p q r s t
^ ^ ^
Since, the transformation matrix (^) T (^) M does not exist. So, we conclude that T is not a linear
transformation. ( Ans. )
7). We have,
3 4 T : R R
And T (^) p q r , , (^) (^) p q ,0, z y ,
So, we must need a 4 3 matrix consisting of real numbers only.
The given transformation can be written as,
p q p
T q
z r
y
^
p q r
p q r
z
y
^
Since, the transformation matrix (^) T (^) M does not exist. So, we conclude that T is not a linear
transformation. ( Ans. )
8). We have,
3 3 T : R R
And T x y z , , (^) (^) 3 x y , 2 x 4 y 3 ,5 z x 4 y 2 z
So, we must need a 3 ^3 matrix consisting of real numbers only.
The given transformation can be written as,
x (^) x y
T y x y z
z (^) x y z
^
x y z
x y z
x y z
^
x
y
z
Mv
Since,
T M
is a matrix of order 3 3 , so we conclude that T is a linear
transformation. ( Ans. )
Now,
T
Since, (^) T (^) 0 , so
1 T
exists.
Hence,
1 T
exists and
1 T
will be the transformation matrix.
The cofactors are, A 11^ 4,^ A 12^ ^ 11,^ A 13^ ^ 12,^ A 21^ ^ 2,^ A 22^ 6,^ A 23^ 7,^ A 31^ ^ 3,^ A 32 9,
33
A 10.
The matrix of cofactors is,
t
adj T
^
1
adj T
T
T
^ ^
