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    furqannoor's Avatar
    furqannoor Posts: 7, Reputation: 1
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    #1

    Dec 8, 2009, 06:40 PM
    vectors maths
    Show that three of the vectors u=(1,1,-1), v=(-1,1,1), w=(1,3,-1), x=(1,1,0) form a basis for space and express the fourth vector as a linear combination of these basis vectors.
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    galactus Posts: 2,271, Reputation: 282
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    #2

    Dec 9, 2009, 07:10 AM
    Show that three of the vectors u=(1,1,-1), v=(-1,1,1), w=(1,3,-1), x=(1,1,0) form a basis for space
    In order to form a basis, they must span and be linearly independent.

    The problem says show that three of the vectors form a basis. Which three? Here are the first three (u,v,w). Try the vector x in there somewhere and see if you can find a basis.

    To show the set spans R^3, we have to show that an arbitrary vector can be expressed as a linear combination



    of the vectors in what we can call S. Expressing this equation in terms of components gives us:





    Equate componets:







    Thus, to show that S spans R^3, we have to show that the system above has a solution for all choices of .

    To prove S is linearly independent, we must show that the only solution of



    is

    So, we have







    has only the trivial solution.

    We can do this both by seeing if the determinant of the system is not equal to 0.



    The determinant is 0, so it does not form a basis.

    Try u,v, and x. I think they form a basis.

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