Page 112 - Advanced Linear Algebra

96 Advanced Linear Algebra

²#³ ²#³
# ~ % b # c % 7  º%» b ker ² ³
6
²%³ ²%³
and so =~ º%» l ker ² . ³

)
For part 4 , if ~ for £ , then ker ² ³ ~ ker ² ³ . Conversely, if

)
2 ~ ker ² ³ ~ ker ² ³, then for % ¤ 2 we have by part 3 ,
=~ º%» l 2

for any . Therefore, if ~ ²%³° ²%³ , it follows that
2
Of course, O ~ O 2
²%³ ~ ²%³ and hence
~ .
Dual Bases
Let be a vector space with basis 8 ~ ¸# “ 0¹ . For each 0 , we can
=

i
define a linear functional # = * by the orthogonality condition

i
#²# ³ ~
Á
is the Kronecker delta function , defined by
where Á
~ if
Á ~ F
£ if
i
i
Then the set 8 ~¸# “ 0¹ is linearly independent, since applying the

equation
i
~ # bÄb # i

gives
to the basis vector #

~ #²# ³ ~ i Á ~
~ ~
for all .

=
Theorem 3.11 Let be a vector space with basis ~ 8 ¸ # “  0 . ¹
i
)
i
1 The set 8 ~¸# “ 0¹ is linearly independent.

)
=
2 If is finite-dimensional, then 8 i is a basis for = i , called the dual basis of
8.
)
Proof. For part 2 , for any  = i , we have
i ²# ³# ²# ³ ~ ²# ³ Á ~ ²# ³

and so ~ ²# ³# i is in the span of 8 i . Hence, 8 * is a basis for = i .