matrix representation of a linear transformation

matrix representation of a linear transformation

Linear transformations

and matrices are the two most fundamental notions in the study of

linear algebra

. The two

concepts

are intimately related. In this article, we will see how the two are related. We assume that all

vector spaces

are finite dimensional and all vectors are written as

column vectors

.

Linear transformations as matrices

Let

V

,

W

be vector spaces (over a common field

k

) of

dimension

n

and

m

respectively. Fix bases

A

=

{

v

1

,

…

,

v

n

}

and

B

=

{

w

1

,

…

,

w

m

}

for

V

and

W

respectively. We shall order these bases so that

v

i

<

v

j

and

w

i

<

w

j

whenever

i

<

j

. To distinguish an ordinary set from an ordered set, we shall adopt the notation

⟨

v

1

,

…

,

v

n

⟩

to mean the set

{

v

1

,

…

,

v

n

}

with

ordering

v

i

≤

v

j

whenever

i

≤

j

. The importance of ordering these bases will be apparent shortly.

For any linear transformation

T

:

V

→

W

, we can write

T

⁢

(

v

j

)

=

∑

i

=

1

m

α

i

⁢

j

⁢

w

i

for each

j

∈

{

1

,

…

,

n

}

and

α

i

⁢

j

∈

k

. We define the

matrix associated with the linear transformation

T

and

ordered bases

A

,

B

by

[

T

]

B

A

:=

(

α

i

⁢

j

)

,

where

1

≤

i

≤

n

and

1

≤

j

≤

m

.

[

T

]

B

A

is a

m

×

n

matrix whose entries are in

k

. When

A

=

B

, we often write

[

T

]

A

:=

[

T

]

A

A

. In

addition

, when both ordered bases are standard bases

E

n

,

E

m

ordered in the

obvious

way, we write

[

T

]

:=

[

T

]

E

m

E

n

.

Examples.

1.

Let

T

:

ℝ

3

→

ℝ

4

be given by

T

⁢

(

x

y

z

)

=

(

x

+

2

⁢

y

+

z

z

-

x

+

y

-

5

⁢

z

3

⁢

x

+

2

⁢

z

)

.

Using the standard ordered bases

E

3

=

⟨

(

1

0

0

)

,

(

0

1

0

)

,

(

0

0

1

)

⟩

⁢

for

⁢

ℝ

3

and

E

4

=

⟨

(

1

0

0

0

)

,

(

0

1

0

0

)

,

(

0

0

1

0

)

,

(

0

0

0

1

)

⟩

⁢

for

⁢

ℝ

4

ordered in the obvious way. Then,

T

⁢

(

1

0

0

)

=

(

1

0

-

1

3

)

,

T

⁢

(

0

1

0

)

=

(

2

0

1

0

)

,

T

⁢

(

0

0

1

)

=

(

1

1

-

5

2

)

,

so the matrix

[

T

]

E

4

E

3

associated with

T

and the standard ordered bases

E

3

and

E

4

is the

4

×

3

matrix

(

1

2

1

0

0

1

-

1

1

-

5

3

0

2

)

.

2.

Let

T

be the same linear transformation as above. However, let

E

3

′

be the same basis as

E

3

except that the order is reversed:

e

3

<

e

2

<

e

1

. Then

[

T

]

E

4

E

3

′

=

(

1

2

1

1

0

0

-

5

1

-

1

2

0

3

)

.

Note that this matrix is just the matrix from the previous example except that the first and the last columns have been switched.

3.

Again, let

T

be the same as before. Now, let

E

4

′

be the ordered basis whose

elements

are those of

E

4

but the order is now given by

e

2

<

e

1

<

e

4

<

e

3

. Then

[

T

]

E

4

′

E

3

′

=

(

1

0

0

1

2

1

2

0

3

-

5

1

-

1

)

.

Note that this matrix is just the matrix from the previous example except that the first two rows and the last two rows have been interchanged.

Remarks

.

•

From the examples above, we note several important features of a

matrix representation

of a linear transformation:

(a)

the matrix depends on the bases given to the vector spaces

(b)

the ordering of a basis is important

(c)

switching the order of a given basis amounts to switching columns and rows of the matrix, essentially multiplying a matrix by a

permutation matrix

.

•

Some basic

properties

of matrix representations of linear transformations are

(a)

If

T

:

V

→

W

is a linear transformation, then

[

r

⁢

T

]

B

A

=

r

⁢

[

T

]

B

A

, where

A

,

B

are ordered bases for

V

,

W

respectively.

(b)

If

S

,

T

:

V

→

W

are linear transformations, then

[

S

+

T

]

B

A

=

[

S

]

B

A

+

[

T

]

B

A

, where

A

and

B

are ordered bases for

V

and

W

respectively.

(c)

If

S

:

U

→

V

and

T

:

V

→

W

, then

[

T

⁢

S

]

C

A

=

[

T

]

C

B

⁢

[

S

]

B

A

, where

A

,

B

,

C

are ordered bases for

U

,

V

,

W

respectively.

(d)

As a result,

T

is

invertible

iff

[

T

]

B

A

is an

invertible matrix

iff

dim

⁡

(

V

)

=

dim

⁡

(

W

)

.

•

We could have represented all vectors as

row vectors

. However, doing so would mean that the matrix representation

M

1

of a linear transformation

T

would be the

transpose

of the matrix representation

M

2

of

T

if the vectors were represented as column vectors:

M

1

=

M

2

T

, and that the application of the matrices to vectors would be from the right of the vectors:

(

a

b

c

)

⁢

(

1

0

-

1

3

2

0

1

0

1

1

-

5

2

)

instead of

(

1

2

1

0

0

1

-

1

1

-

5

3

0

2

)

⁢

(

a

b

c

)

.

Matrices as linear transformations

Every

m

×

n

matrix

A

over a field

k

can be thought of as a linear transformation from

k

n

to

k

m

if we view each vector

v

∈

k

n

as a

n

×

1

matrix (a column) and the

mapping

is done by the

matrix multiplication

A

⁢

v

, which is a

m

×

1

matrix (a column vector in

k

m

). Specifically, we define

T

A

:

k

n

→

k

m

by

T

A

⁢

(

v

)

:=

A

⁢

v

.

It is easy to see that

T

A

is indeed a linear transformation. Furthermore,

[

T

A

]

=

[

T

A

]

E

m

E

n

=

A

, since the

representation

of vectors as

n

-tuples of elements in

k

is the same as expressing each vector under the

standard basis

(ordered) in the vector space

k

n

. Below we list some of the basic properties:

1.

T

r

⁢

A

=

r

⁢

T

A

, for any

r

∈

k

,

2.

T

A

+

T

B

=

T

A

+

B

, where

A

,

B

are

m

×

n

matrices over

k

3.

T

A

∘

T

B

=

T

A

⁢

B

, where

A

is an

m

×

n

matrix and

B

is an

n

×

p

matrix over

k

4.

T

A

is invertible iff

A

is an invertible matrix.

Remark

. As we can see from the discussion above, if we fix sets of base elements for a vector space

V

and

W

, there is a

one-to-one correspondence

between the set of matrices (of the same size) over the underlying field

k

and the set of linear transformations from

V

to

W

.

References

1

Friedberg, Insell, Spence.

Linear Algebra

. Prentice-Hall Inc., 1997.

Title

matrix representation of a linear transformation

Canonical

name

MatrixRepresentationOfALinearTransformation

Date of creation

2013-03-22 17:29:59

Last modified on

2013-03-22 17:29:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A04

Synonym

ordered bases

Synonym

standard ordered bases

Related topic

LinearTransformation

Defines

ordered basis

Defines

matrix representation

Defines

standard ordered basis

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