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Math Help - [SOLVED] Prove polynomials are a generating system (vector spaces)

  1. #1
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    [SOLVED] Prove polynomials are a generating system (vector spaces)

    Let V be the vector space of the polynomials of degree  \leq 2 over \mathbb{R}.

    Prove that the polynomials:

     p_1 = 2T^2 + T + 1
    p_2 = 4T^2 + T
    p_3 = -2T^2 + 2T + 1

    span, or are a generating system of, V.

    If I solve the system of linear equations- how do I prove that they span V?
    Thanks.
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  2. #2
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    Quote Originally Posted by bmp05 View Post
    Let V be the vector space of the polynomials of degree  \leq 2 over \mathbb{R}.

    Prove that the polynomials:

     p_1 = 2T^2 + T + 1
    p_2 = 4T^2 + T
    p_3 = -2T^2 + 2T + 1
    .
    You just need to show these polynomial are linearly independent.
    Say that a_1p_1 + a_2p_2+a_3p_3 = \bold{0} where \bold{0} is the zero-polynomial.
    This means that,
    2a_1+4a_2-2a_3=0
    a_1 +  \ \ \  a_2 + 2a_3 = 0
    a_1 + \ \ \ \ \ \ \ \ \  + a_3 = 0

    Argue that a_1=a_2=a_3=0 is the only solution.
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  3. #3
    Senior Member Twig's Avatar
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    Write the polynomials as vectors in \mathbb{R}^{3} .

     A= \left[\begin{matrix} 1 & 0 & 1\\ 1 & 1 & 2 \\ 2 & 4 & -2 \end{matrix}\right]

    Row operations will show that this matrix A has 3 pivot positions. So it therefore spans  \mathbb{R}^{3} and by the isomorphism between  \mathbb{R}^{3} \mbox{ and } \mathbb{P}_{3} also  \mathbb{P}_{3} .
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