composition of mappings

Pamela Leutwyler

y

x

yx

y

xT

2

0

1

1

0

1T

2

0

1

1

0T

20

01

11

= the matrix for T

y

x

yx

y

xT

2

20

01

11

zy

yx

z

y

x

S3

2

0

2

0

0

1

S

1

1

0

1

0

S

3

0

1

0

0

S

310

012

= the matrix for S

y

x

yx

y

xT

2

20

01

11

zy

yx

z

y

x

S3

2

310

012

R2 R2

R3

T S

ST

v vT )( vTS

y

x

yx

y

xT

2

20

01

11

zy

yx

z

y

x

S3

2

310

012

R2 R2

R3

T S

ST

Find the matrix for ST

y

x

yx

y

xT

2

20

01

11

zy

yx

z

y

x

S3

2

310

012

0

1

1

310

012)___()

0

1

20

01

11

()0

1(

0

1TofcolumnfirstSSTSST

y

x

yx

y

xT

2

20

01

11

zy

yx

z

y

x

S3

2

310

012

0

1

1

310

012)___()

0

1

20

01

11

()0

1(

0

1TofcolumnfirstSSTSST

0

1

1

y

x

yx

y

xT

2

20

01

11

zy

yx

z

y

x

S3

2

310

012

0

1

1

310

012)___()

0

1

20

01

11

()0

1(

0

1TofcolumnfirstSSTSST

y

x

yx

y

xT

2

20

01

11

zy

yx

z

y

x

S3

2

310

012

0

1

1

310

012)___()

0

1

20

01

11

()0

1(

0

1TofcolumnfirstSSTSST

The first column of ST is S(the first column of T)

y

x

yx

y

xT

2

20

01

11

zy

yx

z

y

x

S3

2

310

012

2

0

1

310

012)___(sec)

1

0

20

01

11

()1

0(

1

0TofcolumnondSSTSST

The second column of ST is S(the second column of T)

One way to define the product of two Matrices A and B is:

THE nth COLUMN OF AB IS

A( nth COLUMN OF B )

Because the matrix product AB represents a

COMPOSITION OF MAPPINGS,

It is important that the RANGE OF B is withinThe DOMAIN OF A.

If A is an mn matrix,It has m rows and n columns.

Its domain is Rn, and its range is Rm

If B is an np matrix,It has n rows and p columns.

Its domain is Rp, and its range is Rn

We can form the product AB

mn np

And the answer will be

An mp matrix.

Consider the example:

3234

1125

34

23

12C D CD

32 24 34

Conformable:can be multiplied

Consider the example:

____

____

____

3234

1125

34

23

12C D CD

32 24 34

Consider the example:

____

____

____

3234

1125

34

23

12C D CD

32 24 34

To find the third column of CD take C times the third column of D

Consider the example:

____

____

____

3234

1125

34

23

12C D CD

32 24 34

To break this down further, consider the second entry in column 3

the entry in row 2 column 3 of CD

is row 2 of C DOT column 3 of D

1

In general, for any two conformable matrices A and B:

The entry in row j, column k of AB

Is (row j of A) dot (column k of B)

row j

A B AB

Consider the example:

____

____

____

3234

1125

34

23

12C D CD

64

5

1

2

Consider the example:

____

____

___6

3234

1125

34

23

12C D CD

13

2

1

2

Consider the example:

____

____

__16

3234

1125

34

23

12C D CD

42

1

1

2

Consider the example:

____

____

_416

3234

1125

34

23

12C D CD

13

1

1

2

Consider the example:

____

____

1416

3234

1125

34

23

12C D CD

234

5

2

3

Consider the example:

____

___23

1416

3234

1125

34

23

12C D CD

123

2

2

3

Consider the example:

__18

911223

1416

3234

1125

34

23

12C D CD

102

1

3

4

Sorry – I skipped a few steps!

Consider the example:

51018

911223

1416

3234

1125

34

23

12C D CD