Page 388 - Advanced Linear Algebra

372 Advanced Linear Algebra

)
2 Every ' < n = has an essentially unique expression as a finite sum of
the form
"n &

where & = and the 's are distinct.

"
)
3 Every ' < n = has an essentially unique expression as a finite sum of
the form
%n #

where % < and the 's are distinct.

#
)
4 Every nonzero ' < n = has an expression of the form

'~ % n &
~
%
where the 's are distinct, the 's are distinct and the sets % ¸ ¹ ‹ < and

&
¸& ¹ ‹ = are linearly independent. As to uniqueness, is the rank of ' and

so it is unique. Also, the equation

%n & ~ $n '
~ ~
where the $ 's are distinct, the ' 's are distinct and ¸ $ ¹ ‹ < and
¸' ¹ ‹ = are linearly independent, holds if and only if there exist invertible

!
d matrices ( ~ ² ³ and )~ ² ³ for which ( )~ 0 and
Á
Á

$~ % Á '~ and & Á
~ ~
for ~ Á Ã Á .
Proof. Part 1) merely expresses the fact that ¸" n# ¹ is a basis for < n= .

From part 2), we write
Á " n # ~ @ # A " n " n &
Á ~
Á
Uniqueness follows from Theorem 14.5. Part 3) is proved similarly. As to part
4), we start with the expression from part 2):

"n &

~
&
where we may assume that none of the 's are . If the set ¸ & ¹ is linearly

independent, we are done. If not, then we may suppose (after reindexing if