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Find a basis

  1. Nov 24, 2011 #1
    I don't wan't a solution I wan't only instructions how to solve this problem:

    Find a basis for the span: [tex]\vec{a_{1}}=(1,\,-1,\,6,\,0),\,\vec{a_{2}}=(3,\,-2,\,1,\,4),\,\vec{a_{3}}=(1,\,-2,\,1,\,-2),\,\vec{a_{4}}=(10,\,1,\,7,\,3)[/tex]
     
  2. jcsd
  3. Nov 24, 2011 #2
    Make a matrix with a1, a2, a3, and a4 in separate rows, with each component of each vector in a separate column. Put it in row reduced echelon form. The nonzero rows of your new matrix are the vectors that form the basis.
     
    Last edited: Nov 24, 2011
  4. Nov 25, 2011 #3
    So under your guidance the row reduced eshelon form of the matrix:

    [tex]A=\left( \begin{array}{cccc}1 &-1 & 6 & 0\\ 3 &-2 & 1 & 4\\ 1 &-2 & 1 &-2\\ 10 & 1 & 7 & 3\\ \end{array} \right)[/tex] is [tex]\left( \begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{array} \right)[/tex] so the basis are vectors [tex]\vec{e_1}(1,\,0,\,0,\,0),\,\vec{e_2}(0,\,1,\,0,\,0),\,\vec{e_3}(0,\,0,\,1,\,0)[/tex] and [tex]\vec{e_4}(0,\,0,\,0,\,1)[/tex] right?
     
  5. Nov 25, 2011 #4
    yeah that's what I got
     
  6. Nov 25, 2011 #5
    Thanks a lot for the tips :).
     
  7. Nov 26, 2011 #6

    HallsofIvy

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    Which says that the span of those four vectors is, in fact, all of [itex]R^4[/itex].
     
  8. Nov 26, 2011 #7
    Because the span of those vectors is all of R^4, both the 4 standard R^4 basis vectors and the column vectors of your matrix form a basis. What's more interesting is the case when the column vectors do not span the entire space. Then you take the column vectors from your matrix that correspond to the columns with pivots in reduced form, and those form a basis for the column space.
     
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