Page 122 - Advanced Linear Algebra

106 Advanced Linear Algebra

Corollary 3.20 Let B ²= Á > ³ , where = and > are finite-dimensional.

Then rk²³ ~ rk² d . ³
In the finite-dimensional case, and d can both be represented by matrices.
Let

8 9 ~² Á Ã Á ³ and ~² Á Ã Á ³

be ordered bases for and > , respectively, and let
=
i
i
8 i i i 9 ~² Á Ã Á ³ and i ~² Á Ã Á ³

be the corresponding dual bases. Then
i

²´ µ 89Á ³ Á ~²´ µ ³ ~ ´ µ
9

and
i
i
i
²´ d µ Á 98 i ³ Á ~ ²´ d ² ³µ ³ ~ ´ d ² ³µ ~ i d ² ³² ³ ~ ² ³ i i
i
i
8

Comparing the last two expressions we see that they are the same except that the
roles of and are reversed. Hence, the matrices in question are transposes.

B
Theorem 3.21 Let ²= Á > ³ , where and > are finite-dimensional. If 8

=
=
and are ordered bases for and > , respectively, and 8 i and 9 i are the
9
corresponding dual bases, then
´ d µ 98 i ~ Á ² ´ µ 8 9 ³ Á !
i
In words, the matrices of and its operator adjoint d are transposes of one

another.
Exercises
1. If is infinite-dimensional and is an infinite-dimensional subspace, must
=
:
the dimension of =°: be finite? Explain.
2. Prove the correspondence theorem.
3. Prove the first isomorphism theorem.
4. Complete the proof of Theorem 3.9.
:
=
5. Let be a subspace of . Starting with a basis ¸ ÁÃÁ ¹ for :Á how

would you find a basis for =°: ?
6. Use the first isomorphism theorem to prove the rank-plus-nullity theorem
rk²³ b null²³ ~ dim ²= ³

for B ²= Á > ³ and dim ²= ³  B .
7. Let B ²= ³ and suppose that is a subspace of = . Define a map
:
Z
¢ = °: ¦ = °: by