Subsets of the complex space

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  • #1
daniel_i_l
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Homework Statement


Let:
[tex] \alpha \in C_{4}[x] [/tex] (the space of all 4-deg complex ploymonials)
We'll define:
[tex]
U = Sp( \{\ x^3 - 2i \alpha x, x^2 +1 \} ) \\
W = Sp( \{\ x^3 + ix, (1-i)x^2 - \alpha x \})
[/tex]
as subspaces of [tex]C_{4}[x] [/tex]
a) find all values of alpha so that:
[tex] W \cap U \neq \{ 0 \} [/tex]
b) does:
[tex] C_{4}[x] = W \oplus U [/tex]
when alpha = 0?

Homework Equations


If [tex] C_{4}[x] = W+U [/tex] and [tex] W \cap U = {0} [/tex]
then [tex] C_{4}[x] = W \oplus U [/tex]


The Attempt at a Solution


I'm not sure how to approch this question. Do I find all the cases where
[tex] W \cap U = {0} [/tex] and then write the answer as all the other cases? To do that do I first find all the alphas where u = w for some u in U and for some w in W and then check which one of those forces both u and w to be 0?
And for b, if the answer in a isn't 0 then the answer is no right?
Thanks.
EDIT: Where did all the curly brackets go?
 
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Answers and Replies

  • #2
HallsofIvy
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Homework Statement


Let:
[tex] \alpha \in C_{4}[x] [/tex] (the space of all 4-deg complex ploymonials)
We'll define:
[tex]
U = Sp( \{\ x^3 - 2i \alpha x, x^2 +1 \} ) \\
W = Sp( \{\ x^3 + ix, (1-i)x^2 - \alpha x \})
[/tex]
as subspaces of [tex]C_{4}[x] [/tex]
a) find all values of alpha so that:
[tex] W \cap U \neq \{ 0 \} [/tex]
b) does:
[tex] C_{4}[x] = W \oplus U [/tex]
when alpha = 0?

Homework Equations


If [tex] C_{4}[x] = W+U [/tex] and [tex] W \cap U = {0} [/tex]
then [tex] C_{4}[x] = W \oplus U [/tex]


The Attempt at a Solution


I'm not sure how to approch this question. Do I find all the cases where
[tex] W \cap U = {0} [/tex] and then write the answer as all the other cases? To do that do I first find all the alphas where u = w for some u in U and for some w in W and then check which one of those forces both u and w to be 0?
And for b, if the answer in a isn't 0 then the answer is no right?
Thanks.
EDIT: Where did all the curly brackets go?
For a, I see no reason not to do it directly: any vector in U can be written [itex]m(x^3- 2i\alpha x)+ n(x2+1) and any vector in V can be written [tex]a(x3+ ix)+ b((1+i)x2- \alpha x). Set those equal to each other. For what values of [itex]\alpha[/itex] can you NOT find m,n,a, and b that will work?

As for b, C4[x] includes x4 but neither U nor V does.
 

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