Page 127 - 35Linear Algebra

7.1 Linear Transformations and Matrices 127

n
2
1
The numbers (α , α , . . . , α ) are called the components of the vector v. Two
useful shorthand notations for this are
 1   1 
α α
 α 2   α 2 
v =  .  = (b 1 , b 2 , . . . , b n )  .  .




. .
 .   . 
α n α n
B
7.1.2 From Linear Operators to Matrices
Chapter 6 showed that linear functions are very special kinds of functions;
they are fully specified by their values on any basis for their domain. A
matrix records how a linear operator maps an element of the basis to a sum
of multiples in the target space basis.
More carefully, if L is a linear operator from V to W then the matrix for L
0
in the ordered bases B = (b 1 , b 2 , . . . ) for V and B = (β 1 , β 2 , . . . ) for W, is
j
the array of numbers m specified by
i
1
j
L(b i ) = m β + · · · + m β + · · ·
1
i
i
j
Remark To calculate the matrix of a linear transformation you must compute what
the linear transformation does to every input basis vector and then write the answers
in terms of the output basis vectors:

L(b 1 ), L(b 2 ), . . . , L(b j ), . . .
 1   1   1 
m 1 m 2 m i
 m 2   m 2   m 2 
 2   2   i 

 .   .   . 
= (β 1 , β 2 , . . . , β j , . . .)  .  , (β 1 , β 2 , . . . , β j , . . .)  .  , · · · , (β 1 , β 2 , . . . , β j , . . .)  .  , · · ·
.
.
.
     
 j   j   j 
1 2 i
 m   m   m 
. . .
. . .
. . .
 1 1 1 
m m · · · m · · ·
1 2 i
m m · · · m
 2 2 2 · · · 
 1 2 i 
 . . . 
= (β 1 , β 2 , . . . , β j , . . .)  . . . . . . 
 j j j 
m m · · · m · · ·
 
 1 2 i 
. . . . . . . . .
3
Example 79 Consider L : V → R (as in example 71) defined by
       
1 0 0 0
1
1
1
L   =   , L   =   .
1
0 0 1 0
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