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Math Help - Subspaces

  1. #1
    Junior Member
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    Subspaces

    for n in W, let T_n = {f in F[x] | deg(f) =n or f =0} and P_n = {f in F[x] | deg(f) <= n or f =0}.
    Is T_n a subspace of F[x] over F?
    Is P_n a subspace of F[x] over F?
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  2. #2
    Super Member Gamma's Avatar
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    Iowa City, IA
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    Subspace

    The first is not. consider n=1.

    let f(x)=3x -4, g(x)=-3x +2
    f(x)+g(x)= -2 \not \equiv 0
    so it is not a subspace as it is not closed under addition.

    The second is. You can check. Take arbitrary polynomials of degree n or less and add them, they cannot gain degree. Closed under addition

    Multiply an arbitrary polynomial of degree n or less by a scalar from F, it will still have degree n or less.
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