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Homework Help: Pruning a set of matrices (easy)

  1. May 19, 2012 #1
    http://dl.dropbox.com/u/33103477/prune.png [Broken]

    I am unsure if the the answer is:

    [tex] {\begin{pmatrix}
    2 & 1 \\
    5 & 1
    \end{pmatrix}},
    {\begin{pmatrix}
    3 & -1 \\
    7 & 4
    \end{pmatrix}} [/tex]

    or

    [tex] {\begin{pmatrix}
    2 & 1 \\
    5 & 1
    \end{pmatrix}},
    {\begin{pmatrix}
    3 & -1 \\
    7 & 4
    \end{pmatrix}},
    {\begin{pmatrix}
    2 & 7 \\
    -4 & 1
    \end{pmatrix}} [/tex]

    I'm pretty sure it's the second one but am a bit confused about when the algorithm stops.
    Can someone confirm the answer ?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. May 19, 2012 #2
    I have never encountered the word prune, but by context, I believe it is saying X is linearly dependent and you want to make a set Y (lin ind) from X that spans ##M_{2,2}## as well.

    If that is the case, you just need to verify that none of the matrices in your 3 set are linearly dependent to the other two. If that is the case, then the 3 matrix set is correct.
     
    Last edited by a moderator: May 6, 2017
  4. May 19, 2012 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    Lay them out as 4-dimensional vectors, then look for a subset that generates the whole span. In other words, start with the vectors
    [tex] \begin{array}{l}v_1 = [2,1,5,1] \\v_2 = [3,-1,7,4] \\ v_3 = [5,-5,11,10] \\
    v_4 = [2,7 -4,1]\end{array}[/tex] Obviously, [itex] v_1, v_2[/itex] are linearly independent.
    Are the three vectors [itex] v_1, v_2, v_3[/itex] linearly independent? If not, [itex]v_3[/itex] is in the span of [itex] v_1, v_2.[/itex] If they are linearly independent, throw in [itex] v_4[/itex] and continue the test.

    Note: you can do it all in one step, just by doing row operations on the 4x4 matrix with rows v_i. Basically, you are looking for solutions of the 4x4 linear system [itex] c_1 v_1 + c_2 v_2 + c_3 v_3 + c_4 v_4 = 0.[/itex]

    RGV
     
    Last edited by a moderator: May 6, 2017
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