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Subsets of the complex space

  1. Apr 15, 2007 #1

    daniel_i_l

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    Gold Member

    1. The problem statement, all variables and given/known data
    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?

    2. Relevant 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]


    3. 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?
     
    Last edited by a moderator: Apr 15, 2007
  2. jcsd
  3. Apr 15, 2007 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    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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