Matrices and linear transformations
For grade 1, undergraduate students
Main tasks. In the chapter we see that a linear transformation T: R  R ,
n
n
(or more generally, general linear transformation ) could be represented (that
is, was completely determined by) n2 numbers arranged in a n×n array.
We will bridge a connection between matrices and linear transformations.
Key points.
Linear transformation, linear space, isomorphism.
1. Representing a linear transformation by a matrix
Let us suppose given a linear transformation
T :V W
between the finite-dimensional vector spaces V and W. Let
1 ,  n  be an ordered basis
(that is, a basis whose vectors are placed in a specified order) for V and
ordered basis for W. It is then possible to fine unique numbers aij , i  1,
1, m  an
, n; j  1,
,m
such that
T 1   a111  a212 
 am1m
T 2   a12 1  a212 
 am2 m
T n   a1n 1  a2n 2 
 amn m
which we have seen may be written more compactly as
T  j    aij i
m
j  1, 2,
,n
i 1
The m n matrix A  (aij ) is called the matrix of T relative to the ordered bases
1 ,  n  1, m  .
Remark.
Just as in the case of V  R  W that the columns of A are the coordinates of
3
the vectors T ( j ) relative to the basis 1 ,
m  . Note also that in saying “A is the
matrix of T” , you must also specify relative to what pairs of ordered bases.
EXAMPLE 1.
Let
T : R2  R2
be the linear transformation given by
T ( x, y )  ( y , x )
2
Calculate the matrix of T relative to the standard basis of R .
2
Solution. By the “standard basis of R ”, we of course mean the basis
E1  1,0 , E2   0,1 , and that we are to use this ordered basis in both the domain and
range of T. We have
T 1,0   0,1 , T   0,11,0 ,
so the matrix we seek is
 0 1
A

 1 0
EXAMPLE 2. With T as in Example 1, find the matrix of T relative to the pair of ordered
bases
E E 
1,
2
and
the domain of T and
F F  where
1,
2
F1  (0,1), F2  (1, 0), (Here
 E E  is the basis in
F F  the basis for the range of T. )
1,
2
Solution. We still have
T 1,0   0,1 , T   0,11,0 ,
but now we must write these equation as
T ( E1 )  1F1  0F2 T ( E2 )  0F2  F2
so that
 0 1
B

 1 0
is the matrix that we now seek .
EXAMPLE3. Calculate the matrix of the differentiation operator
D: Pn (R)  Pn (R)

2
relative to the usual basis 1, x, x ,
x n  for Pn (R) .
Solution . We have for m  1, 2,
Dx m  mx m1  0  0 x 
 mx m1   ox n
1,
2
and
D(1)  0.
Thus the matrix that we seek is
0

0
N 

0
0

0

0
.

n
0 
1 0
0 2
0 0
0 0
For example
0
N 
0
0

N  0
0

0

0
N 
0

0
1

0
when n  1
1 0

0 2
when n  2

0 0
1 0 0

0 2 0
when n  3
0 0 3

0 0 0
etc.
Remark. There is of course no reason to restrict us to square matrices. For example we
have the linear transformation
D: Pn (R)  Pn 1 (R)
and we can ask for its matrix relative to the standard bases of Pn (R ) and Pn 1 (R) .
(What size is this matrix? Answer: n  (n  1) ) Computing as before we see the required
matrix is
0 1 0 0


0 0 2 0

N 




0 0 0 n
For example
N   0 1
0
N 
0
0

N  0
0

when n  1
1 0
 when n  2
0 2
1 0 0

0 2 0
when n  3
0 0 3 
etc.
For a related example, try to calculate the matrix of
D: Pn (R)  Pn 1 (R)
((n+1) not n-1). Before you write anything down decide what size it should be.
2.
An isomorphism and its matrix
Proposition. T : V  V is an isomorphism iff its matrix A is invertible (The matrix A is
to be computed relative to any pair
1 n  ,1 n  , of ordered bases for V ).
PROOF. Suppose that T is an isomorphism Let
S :V V
be the linear transformation inverse to T. Let B be the matrix of S relative to the basis
pair
1 n  ,1 n  (Note
that we have interchanged the role of the bases
1 n  ,1 n  ). Thus if B=( bij ) then
S ( j )   bij j . Since the matrix product
AB is the matrix of the linear transformation T S : V  V relative to the basis pair
1 n  , 1 n  .
isomorphisms.
But TS ( )   for all   V
since T and S are inverse
In particular
T  S (  j )  01  0 B2 
and hence the matrix of T  S relative to the bases
1

1
I 


0
0 j 1  1B j  0 j 1
1 n  , 1 n 
 0 n
is
0




1
Therefore AB=I. Likewise, since the matrix product BA is the matrix of the linear
transformation ST : V  V relative to the basis pair-
1  n  , 1  n  .
But
ST (  )   for all  in V because S and T are inverse isomorphisms, and hence as before
we find BA=I. This shows that if
T :V V
is an isomorphism then a matrix S for T is always invertible.
To prove the converse, we suppose that the matrix A of T is invertible.
such that
AB=I=BA.
Let B be a matrix
Let S : V  V be the linear transformation whose matrix relative to the ordered bases
1 n  ,1 n  is B (Note: We have again interchanged the roles of 1 n  , and
1  n 
bases 1
).
Then
 n  , 1
the
matrix
T S :V V
relative
to
the
0

.


1
S  T and I have the same matrix relative to the bases
1  n  so that by
ordered
 n  is
1

1
I 


0
Therefore
of
1  n  ,
S  T  I , that is
S  T (  )   for all  in V
Likewise we see that
TS ( )   for all  in V
so that S and T are inverse isomorphisms.
Remark. In the proof of the last proposition, we used the fact that the matrix of the
transformation I : V  V
defined by I ( )   for all
 relative to the bases
1  n  , 1  n  is the identity matrix . This is not the case if we have two different
bases (or even different orderings on the same basis) in V.
EXAMPLE 4. Find the matrix of the identity linear transformation I : R  R relative
3
to the ordered bases
1  1, 0, 0  ,  2   0,1, 0  ,  3   0, 0,1
1  1,1,1 ,  2  1,1, 0  , 3  1, 0, 0 
Solution. We have
.
3
11  11, 0, 0  ,  1, 0, 0   0 1  0  2  13 ,
1 2   0,1, 0    0,1, 0   01  1 2  (1) 3 ,
1 3   0, 0,1   0, 0,1  11  (1)  2  03 .
So the matrix we seek is
0 0 1 


A   0 1 1 .
 1 1 0 


The moral of the example is that appearances are really deceiving.
Remark. In view of Example above, it is reasonable to expect that when we calculate with
matrices of transformations T : V  V , we insist upon using the same ordered basis
1  n 
twice to do the calculation, rather than work with distinct ordered bases
1  n  , 1 n  .
EXAMPLE 5.
Let T : R  R be the linear transformation given by
3
3
T ( x, y , z )  ( y  z , x  z , y  z ) .
Calculate the matrix of T relative to
(a) the standard basis of R
(b) the basis
3
(1,1,1),(1, 1,0),(1,1, 2) , used twice.
Solution .To do Part (a) we compute as follows:
T : (1, 0, 0)  (0,1,1)
T : (0,1, 0)  (1, 0,1) .
T : (0, 0,1)  (1,1, 0)
Thus the desired matrix is
0 1 1


A  1 0 1.
0 1 0


To so the computations of Part (b) , let us set
1  1,1,1 , 2  1, 1,0 , 3  1,1, 2  .
Then we find
T (1 )  T 1,1,1   2, 2, 2   21  0 2  0 3
T ( 2 )  T 1, 1, 0   1, 1, 0   01   2   3
T ( 3 )  T 1,1, 2   1,1, 2   01  0 2   3 .
So the matrix for Part (b) is
2 0 0 


B   0 1 0  .
 0 0 1


This should serve to illustrate that the matrix of a transformation depends heavily on the bases
we use to compute it.
3. Matrices relative to different bases
Theorem. Let A and B be m n matrices, V and n-dimensional vector space and W an
m-dimensional vector space. Then A and B represent the same linear transformation
T :V W
relative to (perhaps) different pairs of ordered bases iff there exist nonsingular matrices P and
Q such that
A  PBQ1
where P is m n and Q is n  n .
PROOF. There are two things we must prove. First, if A and B represent the same linear
transformation relative to different bases of V and B represent the same linear transformation
relative to different bases of V and W we must construct invertible matrices P and Q such that
A  PBQ1
we must construct a linear transformation
T :V W
and pairs of ordered bases for V and W such that A represents T relative to one pair and S
relative to the other. Consider the first of these. We suppose given bases
1  n  , 1 n  , such that the matrix of T relative to these bases is A, and bases
1 n  , 1 n  such that the matrix of T relative to these bases is B . Let P be the
matrix of
I :W W
relative to the bases
1 n  , 1 n  . Then by the proposition above, P is invertible.
Let Q be the matrix of
I :V V
relative to he bases 1
n  , 1
 n  . Then is also invertible and Q 1 represent the
matrix of
I :V V
relative to the bases
1  n  , 1 n 
Therefore PB is the matrix of T : V  W relative to the bases
1 n  , 1 n  . If
1  n  and
we apply it again we see that PBQ 1 is the matrix of T relative to the bases
1 n  .
1  n  and
But Q is also the matrix of T relative to the bases
1 n  so that
A  PBQ1
as required .
1
To prove the converse, suppose given invertible matrices P and Q such that A  PBQ .
Choose bases
1  n  , 1 n  for V and W respectively . Let
T : V  W be the
linear transformation whose matrix is A relative to these bases. Let
1  P(1 ), ,n  P( n )
1  Q( 1 ), , n  Q(  m )
since P and A are isomorphisms, the collections 1
n  , 1
n  are bases for V and
W respectively. A brute force computation now shows that B is the matrix of T relative to the
bases 1
n  , 1
n  .
EXAMPLE 6. Recall that we are given the linear transformation T : R  R defined
3
3
by
T ( x, y , z )  ( y  z , x  z , y  z )
and
 0 1 1


A  1 0 1
 0 1 0


3
is the matrix of T relative to the standard basis of R , while
2 0 0 


B   0 1 0 
 0 0 1


is the matrix of relative to the ordered basis
(1,1,1),(1, 1,0),(1,1, 2) , of
1
Since there are invertible matrices P, and Q such that A  PBQ
1
R3 .
, our task is to
compute P and
(1)
P
is
Q 1 . We compute them as follows.
the
matrix
(1,1,1),(1, 1,0),(1,1, 2) ,
(2)
Q 1
is
the
matrix
(1,0,0),(0,1,0),(0,0,1) ,
I : R3  R3
of
and
and
of
relative
to
the
basis
pair
the
basis
pair
(1,0,0),(0,1,0),(0,0,1),
I : R3  R 3 relative
to
(1,1,1),(1, 1,0),(1,1, 2).
The computation of P is easy and gives us
1 1 1 


P  1 1 1 
1 0 2 


1
The computation of Q is not hard and depends on the following equations
1
1
1
(1, 0, 0)  (1,1,1)  (1, 1, 0)  (1,1, 2)
3
2
6
1
1
1
(0,1, 0)  (1,1,1)  (1, 1, 0)  (1,1, 2)
3
2
6
1
1
(0, 0,1)  (1,1,1)  0(1, 1, 0)  (1,1, 2)
3
3
so that
1
2

1
1
Q 
3

 1
3
1
2
1

2
0
1 
6 

1 
6 

1
 
3
A tedious computation shows that
A  PBQ1
that is
1

 0 1 1   0 1 1  2 0 0   2

 

 1
1
0
1

1
0
1
0

1
0

 


 0 1 0   0 1 0  0 0 1  3

 

 1

3
1
2
1

2
0
1 
6 

1 
6 

1
 
3
4. Some exercises
1. Find the matrix of following linear transformations relative to the stand are bases for
R3
(a) T : R  R given by
3
5
T  a 1 , a 2 , a 3    a 1 , a 1 a 3 , a 1 a 3 , a 1 a 2  ,
(b) T : R  R
4
4
given by
T  a 1 , a 2 , a 3 , a 4    a 1 , a 1  a 2 , a 1  a 2  a 3 , a 1 a 2 a 3 a 4 
(c) T : R  R given by
3
T  a 1 , a 2 , a 3   a 1 a 2 a 3
(d) T : R  R
2
4
given by
T  a 1 , a 2    a 1 a 2 , 2a 1 ,3a 2 a 1 , 2a 1 a 2 
2. Let T : R  R be the linear transformation whose matrix relative to the standard bases
4
7
is
 1 2 1

 0 1 4
2 0 2

 0 1 3
 3 2 5

0 0 7
4 0 1

0

1
0

1 .
0

1
0 
Find T (1, 2,3, 4) .
3. Let S , T : R  R
3
2
be the linear transformations with matrices
 6 1 2 
 5 0 7 


 and 
 2 1 9 
 2 4 1
respectively. What is the matrix of the linear transformation
3S  7T : R 3  R 2 .
Find (3S-7T)(1,2,3).
4. Let
0 0

a 0
A
b d

c c
0
0
0
f
0

0
0

0
Show that A  0 (Hint: Recall how we did an analogous problem in the 3  3 case using
4
the linear transformation T : R  R
3
3
whose matrix relative to the standard basis was the
analong of A) Can you generalize your result to n  n ?
5. Let T : R  R
2
2
be the linear transformation
T ( x, y )  ( x  3 y,3x  y )
Find the matrix of T relative to
2
(a) the standard basis of R ,
1  (1,1),2  (1, 1) used twice ,
(b) the basis
(c)
1 , 2  in the domain and the standard basis in the range..
6. Let T : R  R
3
3
be the linear transformation
T ( x, y, z )  (3 y  4 z,3x, 4 x) .
Calculate the matrix of T relative to
(a) the standard basis of R
(b) the basis
3
 0, 4, 3 , 5,3, 4 , 5, 3, 4 used twice
7. Find the matrix of the shift S : R  R
n
n
relative to the standard basis used twice.