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

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


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

    $\displaystyle
    p_1 = 1 - x + 2.x^2, p_2 = 3 + x, p_3 = 5 - x + 4x^2, p_4 = -2 - 2x + 2x^2
    $

    The book says it doesn't span $\displaystyle P_2$


    How do they figure out that $\displaystyle P_2$ subspace does not consist of all linear combination of polynomials $\displaystyle v_1, v_2,....,v_r$? 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 $\displaystyle S = \{v_1, v_2,....,v_r\}$ is a set of vectors in a vector space $\displaystyle V$, then the subspace $\displaystyle W$ of $\displaystyle V$ consisting of all linear combinations of the vectors in $\displaystyle S$ is called the space spanned by $\displaystyle v_1, v_2,....,v_r$, and we say that the vectors $\displaystyle v_1, v_2,....,v_r$ span $\displaystyle W$. To indicate that $\displaystyle W$ is the space spanned by the vectors in the set $\displaystyle S = \{v_1, v_2,....,v_r\}$, we write:

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


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

    $\displaystyle
    p_1 = 1 - x + 2.x^2, p_2 = 3 + x, p_3 = 5 - x + 4x^2, p_4 = -2 - 2x + 2x^2
    $

    The book says it doesn't span $\displaystyle P_2$


    How do they figure out that $\displaystyle P_2$ subspace does not consist of all linear combination of polynomials $\displaystyle v_1, v_2,....,v_r$? 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 $\displaystyle 1-x+2x^2\rightarrow (1,-1,2)\,,\,\,3+x\rightarrow (3,1,0)$ , 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.

    $\displaystyle
    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)
    $

    Set up a matrix and solve for $\displaystyle k_1, k_2, k_3$ and $\displaystyle k_4$ 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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