MTH 309 Y
LECTURE 8.1
THU 2014.03.20
Recall:
1) If A is an � × � matrix then the function
TA : R� → R�
defined by TA (v) = Av is called the matrix transformation associated
to A.
2) A function T : R� → R� is a linear transformation if
(i) T (u + v) = T (u) + T (v) for all u� v ∈ R�
(ii) T (�u) = �T (u) for any u ∈ R� and any scalar �.
3) Every matrix transformation is a linear transformation.
4) Every Linear transformation T : R� → R� is a matrix transformation:
T (u) = Au
where
A=
�
T (e1) T (e2) � � � T (e�)
The matrix A is called the standard matrix of T .
�
MTH 309 Y
LECTURE 8.2
THU 2014.03.20
Composition of linear transformations
MTH 309 Y
LECTURE 8.3
THU 2014.03.20
Proposition. If S : R� → R� and T : R� → R� are linear transformations then the composition
T ◦ S : R� → R�
is also a linear transformation.
MTH 309 Y
Problem. Let
LECTURE 8.4
TB : R� → R��
be linear transformations. If
A is the standard matrix of TA
TA : R� → R�
B is the standard matrix of TB
what is the standard matrix of TA ◦ TB ?
THU 2014.03.20
MTH 309 Y
LECTURE 8.5
THU 2014.03.20
Definition. Let A
� and let B be an �� matrix
� be an � � matrix
such that B = v1 v2 � � � v� . The product AB is an � × �
matrix defined by
�
�
AB = Av1 Av2 � � � Av�
Note. If TB : R� → R�, TA : R� → R� are linear transformations,
A is the standard matrix of TA
B is the standard matrix of TB
then AB is the standard matrix of TA ◦ TB :
TA ◦ TB (u) = (AB)u
Note. The product AB is defined only if
(number of columns of A) = (number of rows of B)
A
� �
B
��
� �
AB
MTH 309 Y
Example.
LECTURE 8.6
A=
�
0 1 2
3 4 5
�
�
⎡
THU 2014.03.20
⎤
0 −1 2 1
B=⎣4 5 1 0⎦
1 2 3 1
MTH 309 Y
⎡
LECTURE 8.7
THU 2014.03.20
Another view of matrix multiplication
⎤
⎡
�11 · · · �1�
... ⎦
A = ⎣ ...
��1 · · · ���
⎤
�11 · · · �1�
... ⎦
B = ⎣ ...
��1 · · · ���
⎡
⎤
�11 · · · �1�
... ⎦
AB = ⎣ ...
��1 · · · ���
�
��� = ��1 ��2
⎡
⎤
�1�
⎥
� ⎢
⎢ �2� ⎥
� � � ��� · ⎢ .. ⎥ = ��1�1� + ��2�2� + · · · + ������
⎣ . ⎦
���
MTH 309 Y
Example.
LECTURE 8.8
A=
�
0 1 2
3 4 5
�
�
⎡
THU 2014.03.20
⎤
0 −1 2 1
B=⎣4 5 1 0⎦
1 2 3 1
MTH 309 Y
LECTURE 8.9
THU 2014.03.20
Example.
• Acme Inc. makes two types of widgets: WG1 and WG2.
• Each widget must go through two processes: assembly and testing.
• The number of hours required to complete each process is given
by the matrix
asse test
�
�
Example.
1 of widgets: WG1 and WG2.
WG1 two4 types
• Acme Inc. makes
WG2
7
3
• Each widget must go through two processes: assembly and test• Acmeing.
Inc. has 3 plants: in New York, Texas, and Minnesota.
• rates
The number
hours required
to complete
each process is given
• Hourly
for eachofprocess
(in dollars)
are as follows:
by the matrix
NY MN �
� TX
�
�
10
15
12
asse
4 1
test
15 20 15
7
3
• Acme Inc. has 3 plants: in New York, Texas, and Minnesota.
Problem. What is the cost of producing each type of widgets in
• Hourly rates for each process (in dollars) are as follows:
each factory?
�
10
15
15
20
12
15
�
Problem. What is the cost of producing each type of widgets in
each factory?
MTH 309 Y
1) addition
⎡
LECTURE 8.10
THU 2014.03.20
Other operations on matrices
⎤
⎡
⎤
�11 · · · �1�
�11 · · · �1�
... ⎦ are �×� matrices
... ⎦, B = ⎣ ...
If A = ⎣ ...
��1 · · · ���
��1 · · · ���
then
⎡
⎤
�11 + �11 · · · �1� + �1�
...
...
⎦
A+B = ⎣
�1� + ��1 · · · ��� + ���
Note. The sum A + B is defined only if A and B have the same
dimensions.
MTH 309 Y
LECTURE 8.11
THU 2014.03.20
2) scalar multiplication
⎡
⎤
�11 · · · �1�
... ⎦ and � is a scalar then
If A = ⎣ ...
��1 · · · ���
⎡
⎤
��11 · · · ��1�
.
.
.
.. ⎦
⎣
�A =
.
���1 · · · ����
MTH 309 Y
LECTURE 8.12
THU 2014.03.20
Properties of matrix algebra
1) A(BC ) = (AB)C
2) (A + B)C = AC + BC
A(B + C ) = AB + AC
3) I� = the � × � identity matrix
⎡
⎢
⎢
I� = ⎢
⎣
If A is an � × � matrix then
⎤
1 0 ··· 0
⎥
0 1 ··· 0 ⎥
... ... . . . ... ⎥
⎦
0 0 ··· 1
A · I� = A
I� · A = A
MTH 309 Y
LECTURE 8.13
THU 2014.03.20
Non-commutativity of matrix multiplication
1) If AB is defined then BA need not be defined.
2) Even if both AB and BA are defined then usually
AB �= BA
MTH 309 Y
LECTURE 8.14
THU 2014.03.20
One more operation on matrices: matrix transpose
Definition. The transpose of a matrix A is the matrix AT such that
(rows of AT ) = (columns of A)
Properties of the transpose
1) (AT )T = A
2) (A + B)T = AT + BT
3) (AB)T = BT AT