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Math Help - [Linear Algebra] problem with finding base and dimension

  1. #1
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    [Linear Algebra] problem with finding base and dimension

    Hi,
    I stuck with this question:
    U,W are sub spaces of R4[x]:
    ImageShack - Image Hosting :: 84809558ns8.jpg
    Find base and dimension for:
    U,W,U+W,
    and : U intersection B

    Thanks a lot!
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  2. #2
    MHF Contributor

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    U is the subspace spanned by x^3+ 4x^2- x+ 3, x^3+ 5x^2+ 5, and 3x^3+ 10x^2+ 5. If those three vectors are independent then they form a basis for U themselves.
    Suppose [tex]a(x^3+ 4x^2- x+ 3)+ b(x^3+ 5x^2+ 5)+ c(x^3+ 10x^2+ 5)= 0[/itex]. That gives (a+ b+ c)x^3+ (4a+ 5b+ 10c)x^2+ (-a)x+ (3a+ 5b+ 5c)= 0. Since x^3, x^2, x, and 1 are independent themselves, we must have a+ b+ c= 0, 4a+ 6b+ 10c= 0, -a= 0, and 3a+ 5b+ 5c= 0. From the third equation, a= 0. so the others are b+ c= 0, 6b+ 10c= 0, and 5b+ 5c= 5(b+ c)= 0. The first and last equations are satisfied as long as c= -b but that makes the other equation 6b+ 10(-b)= -4b= 0 so b= 0 and c= 0. Yes, those three vectors are independent and so form a basis for U which is three dimensional.

    Do the same thing with V.
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