Page 82 - Advanced Linear Algebra
P. 82
66 Advanced Linear Algebra
given. If and are given, then there is a unique for which ( c ~ 4 Á 98 and
9
8
(
9
so there is a unique for which (~ 4 89Á .
Theorem 2.13 If we are given any two of the following:
)
1 an invertible d matrix (
)
8
2 an ordered basis for -
)
9
3 an ordered basis for - .
then the third is uniquely determined by the equation
(~ 4 89Á
The Matrix of a Linear Transformation
Let ¢= ¦ > be a linear transformation, where dim ²= ³ ~ and
9
dim²> ³ ~ and let 8 ~ ² Á Ã Á ³ be an ordered basis for = and an
ordered basis for > . Then the map
8 ¢ ´#µ ¦ ´ #µ 9
is a representation of as a linear transformation from - to - , in the sense
(
)
that knowing along with and , of course is equivalent to knowing . Of
8
9
course, this representation depends on the choice of ordered bases and .
8
9
Since is a linear transformation from - to - , it is just multiplication by an
d matrix (, that is,
´#µ ~ (´#µ 8 9
, we get the columns of as follows:
(
Indeed, since ´ µ ~ 8
( ² ³ ~ ( ~ ( ´ # µ ~ ´ 8 µ 9
Theorem 2.14 Let B ²= Á > ³ and let 8 ~ ² Á Ã Á ³ and be ordered
9
bases for and > , respectively. Then can be represented with respect to 8
=
and as matrix multiplication, that is,
9
´#µ ~ ´µ 8 9 , 9 ´#µ 8
where
´µ 89, ~ ²´ µ Ä ´ µ ³ 9 9
is called the matrix of with respect to the bases and . When =~ > and
8
9
8 9 ~ ´ µ, we denote 88 ´ by µ 8 and so
,
´#µ ~ ´µ ´#µ 8 8
8
Example 2.4 Let +¢ F ¦ F be the derivative operator, defined on the vector
space of all polynomials of degree at most . Let ~ ~ 8 ² 9 Á % Á % ³ . Then
