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COMPOSITION OF MAPPINGS. Matrix Multiplication. Pamela Leutwyler. = the matrix for T. = the matrix for S. R 3. R 2. T. S. R 2. S T. R 3. Find the matrix for ST. R 2. T. S. R 2. S T. The first column of ST is S(the first column of T). - PowerPoint PPT Presentation
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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
