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Math Help - Linear Algebra help

  1. #1
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    Linear Algebra help

     {v_{1}, v_{2}, v_{3} } is a basis for a three dimensional real vector space V. Show that the set  {v_{1} + v_{2} , v_{2} + v_{3},  v_{3}+ v_{1} } is also a basis for V.

    I am finding it really hard ot understand the concept of finding bases, and my notes dont make sense, am not even sure where to start, do i first find a spanning set ?
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  2. #2
    Senior Member MacstersUndead's Avatar
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    Re: Linear Algebra help

    Note that if w_1 = v_1 + v_2, w_2 = v_2+v_3, w_3 = v_3+v_1 then

    w_1-w_2+w_3 = 2v_1 and so v_1 = (1/2)(w_1-w_2+w_3)

    The same can be done to find a linear combination of w's for v_2,v_3. Can you finish?
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  3. #3
    Super Member ILikeSerena's Avatar
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    Re: Linear Algebra help

    Hi Tweety!

    To show the new set is a basis for V, you should show that each of the original vectors can be constructed from the new set.
    Does that make sense?
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  4. #4
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    Re: Linear Algebra help

    so some how show  v_{1} =  v_{1} + v_{2} ? Still not sure where to start though
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  5. #5
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    Re: Linear Algebra help

    Quote Originally Posted by MacstersUndead View Post
    Note that if w_1 = v_1 + v_2, w_2 = v_2+v_3, w_3 = v_3+v_1 then

    w_1-w_2+w_3 = 2v_1 and so v_1 = (1/2)(w_1-w_2+w_3)

    The same can be done to find a linear combination of w's for v_2,v_3. Can you finish?
    But thats not a linear combination ?
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  6. #6
    Senior Member MacstersUndead's Avatar
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    Re: Linear Algebra help

    The new set that ILikeSerena is referring to is your new set of w vectors. If you can show that each v vector is a linear combination of w vectors, then you are done. You are done because let t be an arbitrary vector in V. since t is a vector in V, with v's as the basis, t can be expressed as t = av_1 + bv_2 + cv_3. If you can show that v_1,v_2,v_3 are linear combinations of w vectors, then you can replace each v_i with w's.

    I have given you one such combination. (Verify by replacing each w vector by the sum of v vectors and algebra) In this case by simple replacement  t = a[(\frac{1}{2})(w_1-w_2+w_3)] + bv_2 + cv_3

    Combine like terms after you express v_2 and v_3 in terms of w's and observe the arbitrary vector t in the vector space V is a linear combination of w's, hence the set of w vectors is a basis.
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