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Math Help - Why these polynomials p_1, p_2, p_3, p_4 doesn't span space of degree 2 polynomial?

  1. #1
    Senior Member x3bnm's Avatar
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    Why these polynomials p_1, p_2, p_3, p_4 doesn't span space of degree 2 polynomial?

    We know the definition of span:
    If S = \{v_1, v_2,....,v_r\} is a set of vectors in a vector space V, then the subspace W of V consisting of all linear combinations of the vectors in S is called the space spanned by v_1, v_2,....,v_r, and we say that the vectors v_1, v_2,....,v_r span W. To indicate that W is the space spanned by the vectors in the set S = \{v_1, v_2,....,v_r\}, we write:

    W = span(S) or W = span\{v_1, v_2, ....., v_r\}


    I've a math problem and asked to find out whether the polynomials span a space or not. Determine whether the following polynomials span P_2 (the set of degree 2 polynomials).

    <br />
p_1 = 1 - x + 2.x^2, p_2 = 3 + x, p_3 = 5 - x + 4x^2, p_4 = -2 - 2x + 2x^2<br />

    The book says it doesn't span P_2


    How do they figure out that P_2 subspace does not consist of all linear combination of polynomials v_1, v_2,....,v_r? How does one come to this conclusion?
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  2. #2
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    Quote Originally Posted by x3bnm View Post
    We know the definition of span:
    If S = \{v_1, v_2,....,v_r\} is a set of vectors in a vector space V, then the subspace W of V consisting of all linear combinations of the vectors in S is called the space spanned by v_1, v_2,....,v_r, and we say that the vectors v_1, v_2,....,v_r span W. To indicate that W is the space spanned by the vectors in the set S = \{v_1, v_2,....,v_r\}, we write:

    W = span(S) or W = span\{v_1, v_2, ....., v_r\}


    I've a math problem and asked to find out whether the polynomials span a space or not. Determine whether the following polynomials span P_2 (the set of degree 2 polynomials).

    <br />
p_1 = 1 - x + 2.x^2, p_2 = 3 + x, p_3 = 5 - x + 4x^2, p_4 = -2 - 2x + 2x^2<br />

    The book says it doesn't span P_2


    How do they figure out that P_2 subspace does not consist of all linear combination of polynomials v_1, v_2,....,v_r? How does one come to this conclusion?


    You must mean whether they span or not the space of all polynomials of degree 2 or less.

    Just write each given polynomial as a vector , say 1-x+2x^2\rightarrow (1,-1,2)\,,\,\,3+x\rightarrow (3,1,0) , etc.

    Now form a 3 x 4 (or 4 x 3) matrix with the above vectors and check you get a rank 2 matrix, thus 2 vectors out of

    the four are linearly independent and thus they span a 2-dimensional space, which thus

    cannot be all of P_2 since this space'sdimension is 3 .

    Tonio
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  3. #3
    Senior Member x3bnm's Avatar
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    Never mind. I've found the answer.

    <br />
P_2 = k_1(1 - x + 2.x^2) + k_2(3 + x) + k_3(5 - x + 4x^2) + k_4(-2 - 2x + 2x^2)<br />

    Set up a matrix and solve for k_1, k_2, k_3 and k_4 that's it. If you have a valid solution the vectors span and if not vectors doesn't span.
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  4. #4
    Senior Member x3bnm's Avatar
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    Sorry tonio. May be we've posted at the same time. Thanks for help and explanation.
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