Linear Transformations
A function F from the set V to the set W is a rule that assigns to each element x in V exactly one
element F(x) in W.
If V and W are vector spaces , the function F is called a mapping or a transformation from the
vector space V to the vector space W. It is denoted by F: V  W.
In this case, the vector w = F(v) is called the image of the vector v under the transformation F.
Example 1: Let u = (x, y, z) be any vector in  3 . Then
T(u) = (3x  2 y  z, x  y  2 z )
defines a transformation T from  3 to  2 .
Exercise: Find the image of the vector (1, 2, 3) under the transformation in Example 1.
Observe that if
3  2 1 
A
1  2
1
then the transformation T:  3   2 given above, can be written as
T(u) = Au.
In this case, we call the transformation T:  3   2 a matrix transformation, and the 2 x 3
matrix A is called the matrix of the transformation.
Definition: If V and W are vector spaces, then the mapping T: V  W is a linear
transformation provided that
(a) T( u + v) = T(u) + T(v), and
(b) T(cu) = cT(u).
Exercise: Verify that matrix transformation given in Example 1. is a linear transformation.
Exercise: Is the transformation T(x, y, z) = (3x, y 2 , z 3 ) a linear transformation? If so, find the
matrix of the transformation. If not, explain why not.
Theorem 1: If V and W are vector spaces, then the transformation T: V  W is a linear if and
only if
T( au + bv) = aT(u) + bT(v )
for all pairs of vectors u and v in V and all pairs of scalars a and b, i.e. a transformation between
two vector spaces is linear if, and only if, it preserves linear combinations of pairs of vectors.
Theorem 2: The mapping T:  n   m is a linear transformation if and only if it is a matrix
transformation. The matrix A of the transformation T is given by
A  [T (e1 ) T (e2 ) ... T (en )]
where T (e j ) is the image under T of the jth standard unit basis vector e j  (0,...,1,...,0) with 1 in the
j-th position.
Importance of Theorem 2: If the effect of the transformation on each standard unit basis vector
is known, we can apply Theorem 2 to obtain the matrix of the transformation and thereby obtain
a formula for the transformation.
Exercise: Given the linear transformation T:  3   2 with T (e1 )  (1,  1),
T (e2 )  (3, 4), and T (e3 )  (6,0), find T(x, y, z).
Describing a linear transformation T:  2   2 in geometrical terms:
I.
Reflection in the axes
Consider the effect of the transformation T on each standard unit basis vector of  2 given by
T (e1 )  (1, 0), and T (e2 )  (0,  1).
The matrix of the transformation is
1 0
A
,
0  1
and
T(x, y) = (x, -y).
Exercise: Find the image of the vector (2, 3) under the transformation T(x, y) = (x, -y). Draw the
vectors on the same coordinate plane and describe what you observe.
Exercise: Find the matrix of transformation and the transformation that effects reflection in the
y-axis.
Exercise: Find the matrix of transformation and the transformation that effects reflection in the
line y = x.
II.
Rotation
Consider the effect of the transformation T on each standard unit basis vector of  2 given by
T (e1 )  (cos , sin  ), and
T (e2 )  (cos( 

2
), sin(  

2
))  ( sin  , cos ).
Figure:
y
T (e2 )

e2
T (e1 )

x
e1
The matrix of the transformation is
cos
A
sin 
and
 sin  
,
cos 
T ( x, y )  ( x cos  y sin  , x sin   y cos ) .
Exercise: Find the image of the vector (2, 2) under the transformation
T ( x, y )  ( x cos  y sin  , x sin   y cos )
with  
III.

2
. Draw the vectors on the same coordinate plane and describe what you observe.
Expansion(Compression) in the x-direction
Consider the effect of the transformation T on each standard unit basis vector of  2 given by
T (e1 )  (c, 0), and T (e2 )  (0, 1), c  0
The matrix of the transformation is
c
A
0
0
,
1 
and
T(x, y) = (cx, y).



IV.
If c > 1, the transformation is an expansion
If 0 < c < 1, the transformation is a compression
If c = 1, the transformation is the Identity transformation
Shear in the x-direction
Consider the effect of the transformation T on each standard unit basis vector of  2 given by
T (e1 )  (1, 0), and T (e2 )  (c, 1).
The matrix of the transformation is
1
A
0
c
,
1
and
T(x, y) = (x + cy, y).
Exercise: Find the image of the vector (2, 3) under the transformation
T(x, y) = (x + 2y, y).
Draw the vectors on the same coordinate plane and describe what you observe.
Exercise: Determine the transformation for Shear in the y-direction
Theorem: Suppose that the linear transformation T:  2   2 corresponds to a nonsingular
matrix A. Then T is a finite composition of reflections, expansions, compressions, and shears.
Example: Suppose that T:  2   2 is defined by T(x) = Ax where
 2 6
A

1 4 
We reduce A to I as follows:
4
 2 6
1 4 
1
1 4   ( swap r1 and r2 )  2 6  (2r1  r2 )  0  2  (.5r2 ) 






1 4
1 0
0 1  (4r2  r1 )  0 1




The four elementary matrices corresponding to the four row operations used above are:
0
1
0
0 1
1

, E  1  4 .
E1  
, E2  
, E3 
1
4


0
0

1 
1 0
 2 1

2

We know that
E4 E3 E2 E1 A  I
Hence
-1
-1
-1
-1
A  E1 E2 E3 E4
i.e.
0 1
0 1 4
0 1 1
A




1 0 2  0 1
1 0 2