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Math Help - Linearly Dependent

  1. #1
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    Linearly Dependent

    Dear All,
    I am trying to solve these following problem i pick from a book i am confuse as to how to approach the question i will appreciate your help or solution

    Thank you

    Question
    Which of the following sets of vectors in P
    2 are linearly dependent?

    1. 2-x+4x2, 3+6x +2x2, 2+10x - 4x2
    2. 3+x+x2, 2-x+5x2, 4-3x2
    3. 6-x2, 1+x+4x2
    4. 1+3x+3x2,x+4x2,5+6x+3x2,7+2x-x2

    NB:X2 the 2 infront of the x's are superscripts
    Last edited by barhin; May 19th 2011 at 01:16 PM.
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by barhin View Post
    Dear All,
    I am trying to solve these following problem i pick from a book i am confuse as to how to approach the question i will appreciate your help or solution

    Thank you

    Question
    Which of the following sets of vectors in P
    2 are linearly dependent?

    1. 2-x+4x2, 3+6x +2x2, 2+10x - 4x2
    2. 3+x+x2, 2-x+5x2, 4-3x2
    3. 6-x2, 1+x+4x2
    4. 1+3x+3x2,x+4x2,5+6x+3x2,7+2x-x2

    NB:X2 the 2 infront of the x's are superscripts
    \mathbb{P}_2

    is the vector space of all polynomials of degree 2 or less.

    This is isomorphic to

    \mathbb{P}_2 \cong \mathbb{R}

    So

    2-x+4x^2 \iff 2\mathbf{i}-1\mathbf{j}+4\mathbf{k}

    3+6x+2x^2 \iff 3\mathbf{i}+6\mathbf{j}+2\mathbf{k}


    2+10x-4x^2 \iff 2\mathbf{i}+10\mathbf{j}+4\mathbf{k}

    Can you finish from here
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  3. #3
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    Since you are attempting to do a problem involving "dependent" and "independent" vectors you surely must have seen the definition of "dependent" so just check whether those are satisfied.

    Do there exist three non-zero numbers a, b, and c, such that a(2-x+4x^2)+ b(3+6x +2x^2)+ c(2+10x - 4x^2)= 0. One way to do that to collect the coefficients of the same powers and set the coefficients equal to 0 to get three equations in a, b, and c. Another way would be to set x equal to any three numbers you like to get to get three equations.
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  4. #4
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    Dear hallsoflvy,
    Thanks for suggestions but will appreciate if you could throw more light on the results for me or if possible solution to the question.
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  5. #5
    MHF Contributor alexmahone's Avatar
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    For 1) and 2), check if the determinant of the 9 components is zero.

    For 3), if one vector is not a multiple of the other, they are linearly independent.

    For 4), since you have 4 vectors in a 3-dimensional space, they must be linearly dependent.
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