Page 80 - Advanced Linear Algebra
P. 80
64 Advanced Linear Algebra
Linear Transformations from - to -
-
(
Recall that for any d matrix over the multiplication map
( ²#³ ~ (#
B
is a linear transformation. In fact, any linear transformation ²- Á - ³ has
this form, that is, is just multiplication by a matrix, for we have
2 3 2 Ä ~ Ä 3 ² ³ ~
and so ~ ( , where
(~ 2 Ä 3
Theorem 2.10
1 If is an d matrix over then ( ² - ÁB - . ³
)
(
-
)
2 If ²- Á - ³ then ~ ( , where
B
(~ ² Ä ³
The matrix is called the matrix of .
(
Example 2.3 Consider the linear transformation ¢- ¦ - defined by
²%Á&Á'³ ~ ²% c &Á'Á% b & b '³
Then we have, in column form,
%
%
vy v %c & y v c y vy
& ~ ' ~ &
wz w %b&b' z w z wz
'
'
and so the standard matrix of is
v c y
(~
w z
(
If ( C , then since the image of ( is the column space of , we have
Á
²
²
dim ker ( ³ ³ b ² (rk ³ ~ dim - ² ³
This gives the following useful result.
Theorem 2.11 Let be an d matrix over .
(
-
)
1 ( ¢- ¦ - is injective if and only if rk²(³ ~ n.
)
2 ( ¢- ¦ - is surjective if and only if rk²(³ ~ m.
