Page 483 - Advanced Linear Algebra
P. 483
An Introduction to Algebras 467
~ ² ³ ~ b b ~ 6 b 7 b ² c ³
Hence, any + has the form
~ 6 b c ~ b !
7
where ! s and either ~ or . Hence, s but ¤ s . Thus, every
element of is the sum of an element of and an element of the set
+
s
Z
+~ ¸ + ¹
that is, as sets:
+~ s b + Z
Z
Also, s q + ~ ¸ ¹ . To see that + Z is a subspace of , let "Á # + Z . We wish
+
to show that "b# + Z . If # ~ " for some , s then
" b # ~ ² b ³" + . So assume that " and # are linearly independent. Then
Z
" # and are nonzero and so also nonreal.
#
"
Now, "b# and " c# cannot both be real, since then and would be real. We
have seen that
"b# ~ b
and
"c# ~ b
where Á s , at least one of or is nonzero and Á . Then
²" b#³ b ²" c#³ ~ ² b ³ b² b ³
and so
" b # ~ b b b b b
Collecting the real part on one side gives
b ~ " b # c ² b b b ³
s
Á
Now, if we knew that and were linearly independent over we could
conclude that ~ ~ and so
²" b #³ ~ and ²" c #³ ~
which shows that "b# and " c# are in + Z .
To see that ¸Á Á ¹ is linearly independent, it is equivalent to show that
¸"Á#Á ¹ is linearly independent. But if
