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Math Help - Beginning vector subspace question

  1. #1
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    Beginning vector subspace question

    "Determine if the given set is a subspace of Pn for an appropriate value of n. Justify your answers."

    The first problem under these directions is -

    "All polynomials of the form p(t) = at^2, where a is in R."

    I know what the directions say...but I still don't really get exactly what it's asking. The back of the book says "Yes, by Theorem 1, because the set is Span {t^2}."
    Theorem 1 - "If v1, ... , vp are in a vector space V, then Span {v1, ... , vp} is a subspace of V."

    I don't get how this proves the above problem. To determine if something is a subspace, shouldn't I be given a vector? Or set of vectors? How does a polynomial with degree 2 fit into that theorem? Can anyone please explain what I should be trying to find in each problem like the one above or how to go about trying to find it? Any help is appreciated, thanks.
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  2. #2
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    There are several conditions for something to be a space. For instance,
    if A and B are in the space, is A+B in the space. Also, if (e.g.) A is in the space, is 3A in the space?

    To start off, for your case, if p(t) is at^2, and q(t) is bt^2, is p(t)+q(t) = ct^2 for some c?
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  3. #3
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    So...is t supposed to be treated as a vector and t's coefficient a scalar?
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  4. #4
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    You can think of it that way. Alternatively, you can think of t^2 as just something fixed. It really depends on how the problem is set up. From the answer in the book, it doesn't seem like they want t^2 to be variable.
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