What Conditions Allow Nontrivial Intersection of Subspaces U and W in C4[x]?

[daniel_i_l]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

I'm not sure how to approch this question. Do I find all the cases where
$$W \cap U = {0}$$ 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?

 

[HallsofIvy]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

I'm not sure how to approch this question. Do I find all the cases where
$$W \cap U = {0}$$ 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 $\alpha$ can you NOT find m,n,a, and b that will work?<br /> <br /> As for b, C₄[x] includes x⁴ but neither U nor V does.[/tex][/itex]