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Math Help - need help...about vector space and span

  1. #1
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    Smile need help...about vector space and span

    how do i solve this?

    let V be P2(vector space of all polynomials of degree=2 including the zero polynomial)
    S={P1(t), P2(t)}
    P1=t^2+2t+1
    P2=t^2+2
    does S span P2 ?
    THANKS!
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  2. #2
    o_O
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    Consider a general 2nd-degree polynomial a_{1}t^2 + a_{2}t + a_{3}. We must show that it can be written as a linear combination of the vectors in S:
    \begin{array}{rcl}a_{1}t^2 + a_{2}t + a_{3} & = & c_{1}P_{1}(t) + c_{2}P_{2}(t) \qquad \text{For some arbitrary scalar } c_{1}, c_{2} \\ & = & c_{1}(t^2 + 2t + 1) + c_{2}(t^2 + 2) \\ & \vdots & \\ & = & (c_{1}+c_{2})t^2 + (2c_{1})t + (c_{1} + 2c_{2}) \end{array}

    So you can conclude now?
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  3. #3
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    Yes by definition?
    S spans W = {aP1 + bP2} which includes element P2.
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  4. #4
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    Quote Originally Posted by debs19 View Post
    how do i solve this?

    let V be P2(vector space of all polynomials of degree=2 including the zero polynomial)
    S={P1(t), P2(t)}
    P1=t^2+2t+1
    P2=t^2+2
    does S span P2 ?
    THANKS!
    Quote Originally Posted by o_O View Post
    Consider a general 2nd-degree polynomial a_{1}t^2 + a_{2}t + a_{3}. We must show that it can be written as a linear combination of the vectors in S:
    \begin{array}{rcl}a_{1}t^2 + a_{2}t + a_{3} & = & c_{1}P_{1}(t) + c_{2}P_{2}(t) \qquad \text{For some arbitrary scalar } c_{1}, c_{2} \\ & = & c_{1}(t^2 + 2t + 1) + c_{2}(t^2 + 2) \\ & \vdots & \\ & = & (c_{1}+c_{2})t^2 + (2c_{1})t + (c_{1} + 2c_{2}) \end{array}

    So you can conclude now?

    what would be the exact answer?
    do i need to use some variables like a b c and then solve for c1 and c2?
    thanks
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  5. #5
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    All you need for this question is the fact that V has dimension 3, and so cannot be spanned by a set with only 2 elements. I think the question has confused people because you have used P2 for both one of the elements of S and for the set of all polynomials with degree less than or equal to 2
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