- #1

daniel_i_l

Gold Member

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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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