Page 100 - Advanced Linear Algebra
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84 Advanced Linear Algebra
12. Let Á B = ² ³ where is invertible. Show that
rk ² ³ ~ rk ² ³ ~ rk ³
²
13. Let Á B = ² Á > ³ . Show that
rk²b ³ rk² ³ b rk² ³
14. Let be a subspace of = . Show that there is a B ² = ³ for which
:
ker² ³ ~:. Show also that there exists a ²= ³ for which im ² ³ ~:.
B
15. Suppose that Á B = ² . ³
a Show that ~ ) for some ²= ³ if and only if im ² ³ im ² . ³
B
b Show that ~ ) for some ²= ³ if and only if ker ² ³ ker ² . ³
B
B
16. Let dim²= ³ B and suppose that ²= ³ satisfies ~ . Show that
rk
² ³ dim ²= ³.
17. Let be an d matrix over . What is the relationship between the
-
(
linear transformation ( ¢- ¦ - and the system of equations (? ~ ) ?
Use your knowledge of linear transformations to state and prove various
results concerning the system (? ~ ) , especially when ) ~ .
18. Let have basis 8 ~ ¸# ÁÃÁ# ¹ and assume that the base field for =
-
=
has characteristic . Suppose that for each Á we define
B Á ²= ³ by
# if £
Á ²# ³ ~ F
#b # if ~
B
Prove that the Á are invertible and form a basis for ²= ³ .
=
19. Let ²= ³ . If is a -invariant subspace of must there be a subspace
B
:
; = of for which ² : Á ; ³ reduces ?
20. Find an example of a vector space and a proper subspace of for
=
=
:
which = : .
21. Let dim²= ³ B . If , ²= ³ prove that ~ implies that and
B
are invertible and that ~ ² ³ for some polynomial ²%³ -´%µ .
22. Let B ²= . If ³ ~ for all B ²= ³ show that ~ , for some
-, where is the identity map.
=
23. Let be a vector space over a field of characteristic £ - and let and
be projections. Prove the following:
a The difference c is a projection if and only if
)
~ ~
in which case
ker
im²c ³ ~ im² ³ q ker ² ³ and ²c ³ ~ ker ² ³ l im² ³
Hint: is a projection if and only if c is a projection and so c
is a projection if and only if
