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Determining if the following sets span R^3 and creating a basis, wee! matrices

  1. Dec 11, 2005 #1
    Hello everyone, The problem says:Determine which, if, any, of the following sets span R^3?
    b. {[1 1 -1], [-1 1 1]} note: i'm just transposing them, they should be verical.
    c. {[1 1 0], [0 1 1], [1 0 0], [1 0 1]}
    Those are the 2 sets, he showed us how to do them but i got lost on his steps:
    he writes:
    Intially b does not span R^3, which makes sense, because there are only 2 vector sets in b. He then goes on and writes
    [1 1 -1] [-1 1 1] * [x y z];
    and says use the dot product and name the first column which is, [1 1 -1] x1, then the 2nd column [-1 1 1], x2 and use the dot product.
    he comes out with:
    x+y-z = 0;
    -x + y + z = 0;
    he then writes:
    y = a;
    2y = 0;
    y = 0;
    x = z;

    he totally lost me and now your probably lost as well. Anywho from that he got:
    [1 0 1] which says will make b span R^3 and you would get:
    {[1 1 -1], [-1 1 1], [1 0 1]}
    Any ideas on how he got that last vector? Also i know if the dot porduct of 2 vectors is 0, then its linear indepdant.

    He then says, create 3 bases for R^3 using the above sets. he showed us this:
    [1 1 0] [0 1 1] [1 0 0] [1 0 1]
    [1 0 1] = -[1 1 0] + [ 0 1 1] + 2[1 0 0]
    so he said u don't need
    [1 0 1], but he kind of did this by looking at it, is there a systematic approach to solving it? Thanks.
     
    Last edited: Dec 11, 2005
  2. jcsd
  3. Dec 11, 2005 #2

    shmoe

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    {[1 1 -1], [-1 1 1]} is not a basis of R^3 as it has only two vectors. You can check they are linearly independant, so to get a basis for R^3 you can add any vector not in span{[1 1 -1], [-1 1 1]} as this will be an independant set (why?) of 3 vectors in R^3 and hence is a basis.

    To find a vector not in span{[1 1 -1], [-1 1 1]}, he's finding a vector that is orthogonal to every vector in span{[1 1 -1], [-1 1 1]}.
     
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