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Math Help - direct sum vector space

  1. #1
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    direct sum vector space

    Last problem:
    Suppose that V is a vector space over any field F and that U and W are
    subspaces of V.
    Let V=C[X], the vector space of all polynomials over complex numbers. Let U=Span{1+(1+i)X^2, (1+2i)+iX+X^4} and W=Span{X+(2-i)X^2+(3+i)X^3, (2+2i)+(1+i)X+3X^2+(3+i)X^3+2iX^4}. Show that the sum U+W is actually a direct sum and conclude that U+W has a basis with four elements.

    Thank you so much.
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  2. #2
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    Quote Originally Posted by USCGuy View Post
    Last problem:
    Suppose that V is a vector space over any field F and that U and W are
    subspaces of V.
    Let V=C[X], the vector space of all polynomials over complex numbers. Let U=Span{1+(1+i)X^2, (1+2i)+iX+X^4} and W=Span{X+(2-i)X^2+(3+i)X^3, (2+2i)+(1+i)X+3X^2+(3+i)X^3+2iX^4}. Show that the sum U+W is actually a direct sum and conclude that U+W has a basis with four elements.

    Thank you so much.
    Perhaps, the vectors over the complex field,
    (1,0,1+i,0,0),(1+2i,i,0,0,1),(0,1,2-i,3+i,0),(2+2i,1+i,3,3+i,2i)
    Are linearly independent.
    These, vectors correspond to the corresponding coefficients in the polynomials.
    Last edited by ThePerfectHacker; January 20th 2007 at 03:19 PM.
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  3. #3
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    Huh?

    Could you explain it in a little more detail?
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