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Math Help - Spans

  1. #1
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    Spans

    1) Is the polynomial p(x)=1+x+x^2 in span(1-x+2x^2, -1+x^2, -2-x+5x^2)?

    Would you approach this question like this:

    \lambda_1(1-x+2x^2)+\lambda_2(-1+x^2)+\lambda_3(-2-x+5x^2)=1+x+x^2

    And then just find the coefficients? If the coefficients don't exist, then the polynomial is not in the span.

    2) Is S=\{1+x,1-x^2,x+2x^2\} a spanning set for \mathbb{P}_2?

    I really just need a hint for this question on how to start it up.

    What exactly does it mean is it a spanning set for \mathbb{P}_2
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  2. #2
    MHF Contributor

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    Quote Originally Posted by acevipa View Post
    1) Is the polynomial p(x)=1+x+x^2 in span(1-x+2x^2, -1+x^2, -2-x+5x^2)?

    Would you approach this question like this:

    \lambda_1(1-x+2x^2)+\lambda_2(-1+x^2)+\lambda_3(-2-x+5x^2)=1+x+x^2

    And then just find the coefficients? If the coefficients don't exist, then the polynomial is not in the span.
    Yes, that is correct. And you can find the coefficients by equating coefficients of like powers to get three equations or by simply choosing 3 values for x.

    2) Is S=\{1+x,1-x^2,x+2x^2\} a spanning set for \mathbb{P}_2?

    I really just need a hint for this question on how to start it up.

    What exactly does it mean is it a spanning set for \mathbb{P}_2
    A spanning set is a set that spans all of \mathbb{P}_2. If a, b, and c are any numbers, can you find \lambda_a, \lambda_2, and \lambda_3 so that \lambda_1(1-x+2x^2)+\lambda_2(-1+x^2)+\lambda_3(-2-x+5x^2)=a+ bx+ cx^2?

    (If a spanning set is also linearly independent, then it is a basis.)
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