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Math Help - Subspaces

  1. #1
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    Subspaces

    I need to determine whether U is a subspace of V. If it is not a subspace, I need to state a condition that fails and give a counter example showing that that condition fails.

    a) V is the space of all differentiable functions R->R , and U is the set of differentiable functions whose derivative at 0 takes the value 1.

    b) V is the space of all polynomials with real coefficients, viewed as functions R->R, and U is the set of all differentiable functions R->R.

    c) V=R^4, and U= {(a, ab, b, c) belonging to R^4, a,b,c belong to R}

    I know to prove that these are subspaces of V I need to prove that it is closed under scalar multiplication and addition, but I'm not sure how to write them out.
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  2. #2
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    Quote Originally Posted by skittle View Post
    I need to determine whether U is a subspace of V. If it is not a subspace, I need to state a condition that fails and give a counter example showing that that condition fails.

    a) V is the space of all differentiable functions R->R , and U is the set of differentiable functions whose derivative at 0 takes the value 1.

    b) V is the space of all polynomials with real coefficients, viewed as functions R->R, and U is the set of all differentiable functions R->R.

    c) V=R^4, and U= {(a, ab, b, c) belonging to R^4, a,b,c belong to R}

    I know to prove that these are subspaces of V I need to prove that it is closed under scalar multiplication and addition, but I'm not sure how to write them out.
    What have you tried?
    for 1) let f,g \in U what is
    f'(0)+g'(0) and what does this tell you

    for 2 just verify what you need for a subspace a polynomial can be written as
    \displaystyle p(x)=\sum_{k=0}^{n}a_kx^k

    for 3 try adding two vectors of that form and see if there result is of the correct form.
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