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Linear Algebra: Using Gaussian elimination to find linear independence in vectors

  1. Feb 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Are the vectors a = [1 -1 0 1], b = [1 0 0 1] and c =
    [0 -1 0 1] linearly independent?


    3. The attempt at a solution
    I am mainly confused about whether or not I should have my matrix in row or column form to solve this:

    r 1 -1 0 1
    s 1 0 0 1
    t 0 -1 0 1

    or
    r s t
    1 1 0
    -1 0 -1
    0 0 0
    1 1 1
     
  2. jcsd
  3. Feb 2, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The standard method is to write the matrix having the vectors as columns. However, that is only a convention. Doing it the other way should give the same result- the vectors will be independent, doing this the first way, if you do NOT get a row of all 0s. Doing it writing the vectors as rows, the vectors will be independent if you do NOT get a column of all 0s.

    However, if you have trouble remembering this, it appears you are trying to memorize a method that is, to you, arbitrary. Personally, I prefer to use the definition of "linear independent" which you should have learned anyway. A set of vectors is "linearly independent" if and only if the only linear combination that is equal to the 0 vector has all coefficients equal to 0. That is, in this case, if a<1, -1, 0, 1>+ b<1, 0, 0, 1>+ c<0, -1, 0, 1>= <0, 0, 0, 0> then we must have a= b= c= 0. Is that true?

    Of course, that gives <a+ b, -a- c, 0, a+ b+ c>= <0, 0, 0, 0> or the three equations a+ b= 0, -a-c= 0, a+b+ c= 0. From the first b= -a and from the second, c= -a. Put those into the third equation and solve for a.
     
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