Page 94 - Advanced Linear Algebra

78 Advanced Linear Algebra

which is equivalent to
~c
Of course, this holds if ~ ~ , that is, if ž . But the converse is also
true, provided that char²-³ £ . To see this, we simply evaluate in two
ways:
² ³ ~ c ² ³ ~ c

and
~ ³ c ~ ³ c
²
²
Hence, ~ ~ c and so ~ . It follows that ~ c ~ and so
ž ² . Thus, for char - ³ £ , we have b is a projection if and only if
ž .

Now suppose that b is a projection. For the kernel of b , note that

² b ³# ~ ¬ ² b ³#~ ¬ #~

b ³‹ ker
and similarly, . Hence, ker ² #~ ² ³ q ker ² ³ . But the reverse

inclusion is obvious and so
ker²b ³ ~ ker² ³ q ker² ³

As to the image of b , we have

# im ² b ³ ¬ #~² b ³#~ # b # im ² ³ b im ² ³

and so im²b ³ ‹ im² ³ b im² ³ . For the reverse inclusion, if # ~ % b & ,

then

² b³# ~ ² b³² % b&³ ~ % b& ~ #

and so # im ² b ³ . Thus, im ² b ³ ~ im ² ³ b im ² ³ . Finally, ~

implies that im²³ ‹ ker ² ³ and so the sum is direct and

im²b ³ ~ im² ³ l im² ³

The following theorem also describes the situation for the difference and
product. Proof in these cases is left for the exercises.
=
Theorem 2.26 Let be a vector space over a field of characteristic £ - and
let and be projections.

1 The sum b ) is a projection if and only if ž , in which case
ker

im²b ³ ~ im² ³ l im² ³ and ²b ³ ~ ker ² ³ q ker ² ³
)

2 The difference c is a projection if and only if
~ ~