Orthogonal Diagonalization

Suppose two linear transformations act in the same way on ~x for all vectors. Then we say that these transformations are equal.

Definition 5.15: Equal Transformations

Let S and T be linear transformations from Rn to Rm. Then S = T if and only if for every~x ∈ Rn,

S (~x) = T (~x)

Suppose two linear transformations act on the same vector ~x, first the transformation T and then a second transformation given by S. We can find the composite transformation that results from applying

both transformations.

Definition 5.16: Composition of Linear Transformations

Let T : Rk 7→ Rn and S : Rn 7→ Rm be linear transformations. Then the composite of S and T is

S ◦T : Rk 7→Rm

The action of S ◦T is given by

(S ◦T )(~x) = S(T (~x)) for all~x ∈ Rk

Notice that the resulting vector will be in Rm. Be careful to observe the order of transformations. We

write S ◦T but apply the transformation T first, followed by S.

Theorem 5.17: Composition of Transformations

Let T : Rk 7→ Rn and S : Rn 7→ Rm be linear transformations such that T is induced by the matrix A and S is induced by the matrix B. Then S ◦T is a linear transformation which is induced by the matrix BA.

Consider the following example.

284 Linear Transformations

Example 5.18: Composition of Transformations

Let T be a linear transformation induced by the matrix

A =

[ 1 2

2 0

]

and S a linear transformation induced by the matrix

B =

[ 2 3

0 1

]

Find the matrix of the composite transformation S ◦T . Then, find (S ◦T )(~x) for~x = [

1

4

] .

Solution. By Theorem 5.17, the matrix of S ◦T is given by BA.

BA =

[ 2 3

0 1

][ 1 2

2 0

] =

[ 8 4

2 0

]

To find (S ◦T )(~x), multiply~x by BA as follows [

8 4

2 0

][ 1

4

] =

[ 24

2

]

To check, first determine T (~x): [ 1 2

2 0

][ 1

4

] =

[ 9

2

]

Then, compute S(T (~x)) as follows: [

2 3

0 1

][ 9

2

] =

[ 24

2

]

Consider a composite transformation S ◦T , and suppose that this transformation acted such that (S ◦ T )(~x) =~x. That is, the transformation S took the vector T (~x) and returned it to~x. In this case, S and T are inverses of each other. Consider the following definition.

Definition 5.19: Inverse of a Transformation

Let T : Rn 7→ Rn and S : Rn 7→ Rn be linear transformations. Suppose that for each~x ∈ Rn,

(S ◦T )(~x) =~x

and

(T ◦S)(~x) =~x Then, S is called an inverse of T and T is called an inverse of S. Geometrically, they reverse the

action of each other.

5.3. Properties of Linear Transformations 285

The following theorem is crucial, as it claims that the above inverse transformations are unique.

Theorem 5.20: Inverse of a Transformation

Let T : Rn 7→ Rn be a linear transformation induced by the matrix A. Then T has an inverse trans- formation if and only if the matrix A is invertible. In this case, the inverse transformation is unique

and denoted T−1 : Rn 7→ Rn. T−1 is induced by the matrix A−1.

Consider the following example.

Example 5.21: Inverse of a Transformation

Let T : R2 7→ R2 be a linear transformation induced by the matrix

A =

[ 2 3

3 4

]

Show that T−1 exists and find the matrix B which it is induced by.

Solution. Since the matrix A is invertible, it follows that the transformation T is invertible. Therefore, T−1

exists.

You can verify that A−1 is given by:

A−1 =

[ −4 3

3 −2

]

Therefore the linear transformation T−1 is induced by the matrix A−1. ♠

Exercises

Exercise 5.3.1 Show that if a function T : Rn →Rm is linear, then it is always the case that T ( ~0 ) =~0.

Exercise 5.3.2 Let T be a linear transformation induced by the matrix A =

[ 3 1

−1 2

] and S a linear

transformation induced by B =

[ 0 −2 4 2

] . Find matrix of S ◦T and find (S ◦T )(~x) for~x =

[ 2

−1

] .

Exercise 5.3.3 Let T be a linear transformation and suppose T

([ 1

−4

]) =

[ 2

−3

] . Suppose S is a

linear transformation induced by the matrix B =

[ 1 2

−1 3

] . Find (S ◦T )(~x) for~x =

[ 1

−4

] .

286 Linear Transformations

Exercise 5.3.4 Let T be a linear transformation induced by the matrix A =

[ 2 3

1 1

] and S a linear

transformation induced by B =

[ −1 3

1 −2

] . Find matrix of S ◦T and find (S ◦T )(~x) for~x =

[ 5

6

] .

Exercise 5.3.5 Let T be a linear transformation induced by the matrix A =

[ 2 1

5 2

] . Find the matrix of

T−1.

Exercise 5.3.6 Let T be a linear transformation induced by the matrix A =

[ 4 −3 2 −2

] . Find the matrix

of T−1.

Exercise 5.3.7 Let T be a linear transformation and suppose T

([ 1

2

]) =

[ 9

8

] , T

([ 0

−1

]) =

[ −4 −3

] . Find the matrix of