Page 123 - Advanced Linear Algebra
P. 123
The Isomorphism Theorems 107
Z ²# b :³ ~ # b :
When is well-defined? If is well-defined, is it a linear transformation?
Z
Z
Z
Z
What are im²³ and ker ²³ ?
8. Show that for any nonzero vector #= , there exists a linear functional
= for which ²#³ £ .
i
9. Show that a vector #= is zero if and only if ²#³~ for all = i .
10. Let be a proper subspace of a finite-dimensional vector space and let
:
=
i
#= ± :. Show that there is a linear functional = for which
²#³ ~ and ² ³ ~ for all :.
11. Find a vector space and decompositions
=
=~ ( l ) ~ * l +
with ( * but ) + . Hence, ( * does not imply that ( * .
°
12. Find isomorphic vectors spaces and > with
=
=~ : l ) and > ~ : l +
but ) + . Hence, = > does not imply that = °: > °: .
°
13. Let be a vector space with
=
= ~ :l ; ~ :l ;
=
Prove that if : and : have finite codimension in , then so does : q :
and
codim²: q : ³ dim ²; ³ b dim ²; ³
14. Let be a vector space with
=
= ~ :l ; ~ :l ;
Suppose that : and : have finite codimension. Hence, by the previous
exercise, so does :q : . Find a direct sum decomposition = ~ > l ?
for which 1 > () has finite codimension, 2 > () : q : and 3()
.
? ; b ;
15. Let be a basis for an infinite-dimensional vector space and define, for
8
=
Z
Z
all 8 , the map = i by ² ³ ~ if ~ and otherwise, for all
8. Does
i
Z
8 ¸ ¹ form a basis for = ? What do you conclude about
the concept of a dual basis?
=
16. Prove that if and are subspaces of , then : ² l ; ³ i : i ^ ; i .
:
;
d
17. Prove that ~ and d ~ where is the zero linear operator and is
the identity.
:
18. Let be a subspace of . Prove that = = ² ° : ³ i : .
19. Verify that
d
a) ²b ³ ~ d b d for Á ²= Á > . ³
B
b ² ³ ~ d for any - and ²= Á > ³
d
)
B
20. Let B ²= Á > ³ , where = and > are finite-dimensional. Prove that
rk²³ ~ rk² d ³.
