Sections 1.8/1.9:
Linear Transformations

Recall that the difference between the matrix equation
Ax  b
and the associated vector equation
x1a1  x 2a 2 L  x p a p  b is just a matter of notation.
However the matrix equation Ax  b can arise is linear
algebra (and applications) in a way that is not directly
connected with linear combinations of vectors.

This happens when we think of a matrix A as an object that
acts on a vector x by multiplication to produce a new vector
Ax
Example:
x =
A
  2 3 0 0
 0 2  1 1


2 4

b
2
-2 3 0  0 1
1 
1  0  0   
   2
0  2 1
 1 2 
0 
 
0 
R4
R2
Recall that Ax is only defined if the number of columns
of A equals the number of elements in the vector x.
2 4
R2
  2 3 0 0  2 
 0 2  1 1 1   Undefined

  

3
R
2 4
2
 2 3 0 0    Undefined
 0 2  1 1  1 


 0 
 
2 4
R4
R2
1 
 2 3 0 0  2    1
 0 2  1 1
  
2

 0 
 
 
0 
A
b
x
R
R2
4



So multiplication by A transforms
x into b .
In the previous example, solving the equation Ax = b can be
thought of as finding all vectors x in R4 that are transformed
into the vector b in R2 under the “action” of multiplication by
A.
Transformation:
Any function or mapping
T : R R
n
m
T

Range
Domain
R
Codomain
n
R
m
Matrix Transformation:
Let A be an mxn matrix.
Ax  b
Domain
R
n
x

a11 a12 ...

a21 a22 ...
 ... ......

am1 am2 ...
A
b
Codomain
Rm
x b
A

a1n  x1   a11 x1  a12 x 2  ... a1n x n 
   

a2n  x 2  a21 x1  a22 x 2  ... a2n x n 



...  ...  
...
    

amn  x n  am1 x1  am 2 x 2  ... amn x n 
Example: The transformation T is defined by T(x)=Ax where
T : R R
n
m
For each of the following determine m and n.
1  3 
1. A   3 5 
  1 7 
1 0 0
2. A  

0
1
0


1 3
3. A  

0
1


Matrix Transformation:
Ax=b
A
x
b
m n
Domain
R
n
Codomain
R
m
Linear Transformation:
Definition:
A transformation T is linear if
(i) T(u+v)=T(u)+T(v) for all u, v in the domain of T:
(ii) T(cu)=cT(u) for all u and all scalars c.
Theorem: If T is a linear transformation, then
T(0)=0 and
T(cu+dv)=cT(u)+dT(v) for all u, v and all scalars c, d.
Example. Suppose T is a linear transformation from R2 to R2
 1    2
 0   0 


such that T        and T        . With no additional
 0   1
 1   1 
information, find a formula for the image of an arbitrary x in R2.
x1 
1 
0
x     x1    x2  
x 2 
0
1 

 1 
0  

 T x  T  x1    x2   
1  
 0 
 1  
 0  
 x1T      x2T    
 0  
 1  
 2
0
 x1    x2  
 1
1 
  x1    2 0  x1 
 T      
 

  x2     1 1   x2 
 1    2
T       
 0   1
 0   0 
T       
 1   1 
  x1    2 0  x1 
 T      
 

  x2     1 1   x2 
 4  2 0 2
1   1 1  1 
  
  
Theorem 10.
Let T : R n R m be a linear transformation. Then there exists a
unique matrix A such that T x  Ax for all x in Rn.
In fact, A is the m n matrix whose jth column is the vector T (e j )
where e j is the jth column of the identity matrix in Rn.

 1    2
T       
 0   1
 0   0 
T       
 1   1 
3
 1    
T      1 
 0  5
 
 2
 0    
T       1
 1    5 
 
  x1    2 0  x1 
 T      
 

  x2     1 1   x2 
  x1  
 T    
  x2  
3
 1

5
2
 x1 

1   
  x2 
5
A is the standard matrix for the linear transformation T
Find the standard matrix of each of the following transformations.
Reflection through
the x-axis
1
0

Reflection through
the y-axis
  1 0
 0 1


Reflection through
the y=x
0
1

0
1
1
0
Reflection through
the y=-x
 0  1
 1

0


Reflection through
the origin
0
 1
 0  1


Find the standard matrix of each of the following transformations.
Horizontal
Contraction &
Expansion
k
k
Vertical
Contraction &
Expansion
k
0

0
1
1
 0

0
k 
Projection onto
the x-axis
1
0

0
0
Projection onto
the y-axis
0
0

0
1 
Applets for transformations in R2
From Marc Renault’s collection…
Transformation of Points
http://webspace.ship.edu/msrenault/ggb/linear_transformations_points.html
Visualizing Linear Transformations
http://webspace.ship.edu/msrenault/ggb/visualizing_linear_transformations.html
Definition
m
A mapping T : R n  R m is said to be onto R
n
if each b in R m is the image of at least one x in R .
Definition
A mapping T : R n  R m is said to be one-to-one
n
m
R
R
if each b in
is the image of at most one x in
.
Theorem 11
Let T : R n  R m be a linear transformation. Then,
T is one-to-one iff T ( x)  0 has only the trivial solution.
.
Theorem 12
Let T : R n  R m be a linear transformation with standard
matrix A.
m
1. T is onto iff the columns of A span R .
2. T is one-to-one iff the columns of A are linearly independent