EDUCATIVA INTEGRAL
3.5. EL CURRÍCULO PARA EL ENFOQUE COMPETENCIAL
The operations are so far very natural and easy. Multiplication of two matrices is more complicated, but a simple example illustrates that there is one useful natural definition.
Suppose that we wish to change variables; that is, we wish to work with new variables y1 and Y2 that are defined in terms of the original variables x1 and x2 by the equations
Y1 = a11x1 + a12X2 Y2 = a21 X1 + a22x2
This is a system of linear equations like those in Chapter 2.
It is convenient to w1ite these equations in matrix form. Let
x
=[�J y
=[��].
and
A
=[ai
1 a12]
be the coefficient matrix. Then the change of variables equations a21 a22can be written in the form
y
=Ax,
provided that we define theproduct of A and x
according to the following rule:
(3.1)
It is instructive to rewrite these entries in the right-hand matrix as dot products. Let a1 =
[ ::� ]
and a2 =[ :�� ]
so that af
=[
a11 a12]
anda{
=[
a21 a22]
are the rows ofA.
Then the equation becomesThus, in order for the right-hand side of the original equations to be represented cor
rectly by the matrix product
Ax,
the entry in the first row ofAx
must be the dot product of the first row ofA
(as a column vector) with the vectorx;
the entry in the second row must be the dot product of the second row ofA
(as a column vector) withx.
Suppose there is a second change of variables from
y
to z:Z1 = b11Y1 + b12Y2 Z2 = b21Y1 + b22Y2
EXAMPLE7
Definition
Matrix Multiplication
In matrix form, this is written z = By. Now suppose that these changes are performed one after the other. The values for y1 and Y2 from the first change of variables are substituted into the second pair of equations:
ZJ = b11(a11X1 + a12X2) + b12(a21X1 + a22X2) z2 = b21(a11x1 + a12x2) + b22(a21X1 + a22x2) After simplification, this can be written as
z=
[
b11a11 + b12a21b21a11 + b22a21
b11a12 + b12a22
]
xb21a12 + b22a22
We want this to be equivalent to z = By = BAx. Therefore, the product BA must be b11a12 + b12a22
]
b21a12 + b22a22 (3.2)
Thus, the product BA must be defined by the following rules:
• (BA)11 is the dot product of the first row of Band the first column of A.
• (BA)12 is the dot product of the first row of Band the second column of A.
• (BA)ii is the dot product of the second row of Band the first column of A.
• (BA)22 is the dot product of the second row of B and the second column of A.
[
2 3] [
5 1] [
2(5) + 3(-2) 2(1) + 3(7)] [
4 23]
4 1 -2 7 - 4(5) + 1(-2) 4(1) + 1(7) - 18 11
We now want to generalize matrix multiplication. It will be convenient to use
b[
to represent the i-th row of B and a1 to represent the }-th column of A. Observe from our work above that we want the ij-th entry of BA to be the dot product of the i-th row of Band the }-th column of A. However, for this to be defined,
bj
must have the same number of entries as a1. Hence, the number of entries in the rows of the matrix B (that is, the number of columns of B) must be equal to the number of entries in the columns of A (that is, the number of rows of A). We can now make a precise definition.Let B be an m x n matrix with rows
bf,
. . ., b�
and A be an n x p matrix with columnsa1, • • • , ap. Then, we define BA to be the matrix whose ij-th entry is
Remark
-; -;
(BA)iJ = b; · ai
If B is an m x n matrix and A is a p x q matrix, then BA is defined only if n = p.
Moreover, if n = p, then the resulting matrix ism x q.
More simply stated, multiplication of two matrices can be performed only if the number of columns in the first matrix is equal the number of rows in the second.
EXAMPLES
EXAMPLE9
EXERCISE4
EXAMPLE 10
Perform the following operations.
3 1
[
2 3 0 l]
1 2(a) 4 -1 2 -1 2 3
Solution:
[�
0 s
3 1 3 0 1
)
1 2-1 2 -1 2 3 0 s
(b)
H -� m� !l
Section 3.1 Operations on Matrices 123
=
[
15 9 133]
Solution:
[-�
-l 0 1 2 s�i [� �i
=[�� ��i
2 s[
2 1 _ 33] [ � �
2 -3]
is not defined because the first matrix has two columns and the sec-ond matrix has three rows.Let A =
[ � � -�]
and B =[ � �].
Calculate the following or explain why they are not defined.(a) AB (b) BA (c) AT A (d) BET
LetA =
m
andB =m
Compu�ATBSolution: AT B =
[I
2 3] [�]
= [1(6) + 2(5) + 3(4)] = [28]Obsme that if we let X =
m
and'! =m
be vectors in 1<3, then X ·y
= 28, This matches the result in Example 10. This result should not be surprising since wehave defined matrix multiplication in terms of the dot product. More generally, for any
x,y
E JRn, we haveXTy
=X
·y
where we interpret the 1 x 1 matrix on the right-hand side as a scalar. This formula will be used frequently later in the book.
Defining matrix multiplication with the dot product fits our view that the rows of the coefficient matrix of a system of linear equations are the coefficients from each equation. We now look at how we could define matrix multiplication by using our alternate view of the coefficient matrix; in that case, the columns of the coefficient matrix are the coefficients of each variable.
Observe that we can write equations (3. 1 ) as
Ax=
[
a11 x1 + a12x2a11 X1 + a12X2]
=[
a11a11]
xi +[
a12a12J
x2That is, we can view Ax as giving a linear combination of the columns of A. So, for an m x n matrix A with columns a1, ... , a11 and vector
x
E JR", we haveXt
j
an
] :
= X1a1 + ·'' + Xnan XnUsing this, observe that (3.2) can be written as BA =
[
Ba1 Ba2]
Hence, in general, if A is an m x n matrix and B is a p x m matrix, then BA is the p x n matrix given by
(3.3) Both interpretations of matrix multiplication will be very useful, so it is important to know and understand both of them.
Remark
We now see that linear combinations of vectors (and hence concepts such as spanning and linear independence), solving systems of Linear equations, and matrix multiplica
tion are all closely tied together. We will continue to see these connections later in this chapter and throughout the book.