文档介绍：Chapter Six
Linear Functions and Matrices
Matrices
n p
Suppose f :R ® R be a linear function. Let e1 ,e2 ,K,en be the coordinate
n n
vectors for R . For any x Î R , we have x = x1e1 + x2e2 +K+xnen . Thus
f ( x) = f (x1e1 + x2 e2 +K+xnen ) = x1 f (e1 ) + x2 f (e2 )+K+xn f (en ) .
Meditate on this; it says that a linear function is entirely determined by its values
f (e1 ), f (e2 ),K, f (en ). Specifically, suppose
f (e1 ) = (a11 ,a21 ,K,a p1 ),
f (e2 ) = (a12 ,a 22 ,K,a p2 ),
M
f (en ) = (a1n ,a 2n ,K,a pn ).
Then
f ( x) = (a11 x1 + a12 x2 +K+a1n xn , a 21 x1 + a 22 x2 +K+a 2n xn ,K,
a p1x1 + a p2 x2 +K+a pn xn ).
The numbers aij thus tell us everything about the linear function f. . To avoid labeling
these numbers, we arrange them in a rectangular array, called a matrix:
éa11 a12 K a1n ù
êa a a ú
ê 21 22 K 2n ú
ê M M ú
ê ú
ëa p1 a p2 K a pn û
The matrix is said to represent the linear function f.
For example, suppose f : R 2 ® R 3 is given by the receipt
f ( x1 ,x2 ) = (2x1 - x2 , x1 + 5x2 , 3x1 - 2x2 ) .
Then f (e1 ) = f (10, ) = (213, , ) , and f (e2 ) = f (01, ) =(-15, ,- 2) . The matrix representing f
is thus
é2 - 1ù
ê ú
ê1 5 ú
ëê3 - 2ûú
Given the matrix of a linear function, we can use the matrix pute f ( x) for
any x. This calculation is systematized by introducing an arithmetic of matrices. First,
we ne