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Determining the rank of a matrix

  1. Dec 28, 2016 #1
    1. The problem statement, all variables and given/known data
    2136c8.png

    2. Relevant equations
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    3. The attempt at a solution
    I know that they got a rank of 2 since there are 2 linearly independent columns but what if we decided to count rows? In that case we would have 4 linearly independent rows which would suggest the rank is 4? How do we reconcile between these?
     
  2. jcsd
  3. Dec 28, 2016 #2

    fresh_42

    Staff: Mentor

    You should try to prove the linear independence of the four rows. Or simply draw them in a plane.
     
  4. Dec 28, 2016 #3

    Ray Vickson

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    No. Just because you have 4 rows does not mean you have 4 independent rows. In fact, you can have at most 2 independent vectors of dimension 2!

    In fact, you can see that Row_3 = 2*Row_2 - Row_1 and Row_4 = - Row_1 - 4*Row_2..

    REMEMBER: in any matrix, the number of linearly-independent rows equals the number of linearly-independent columns. That is an important basic theorem of linear algebra.
     
  5. Dec 28, 2016 #4

    Ray Vickson

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    I hope you mean linear dependence.
     
  6. Dec 28, 2016 #5

    fresh_42

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    No, I meant linear independency. My goal was to let the OP see by himself why this won't be possible and learn out of it.
     
  7. Dec 28, 2016 #6

    Ray Vickson

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    OK: I found the wording confusing.
     
  8. Dec 28, 2016 #7

    fresh_42

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    He claimed the rows to be linear independent, so to ask for a proof seemed to be logic. Thus he would have learnt multiple things: don't claim, what you cannot prove, the calculations with linear dependencies and linear independencies are basically the same, and a (failed) proof would have simultaneously shown two free parameters solving the dimension formula. All within this tiny example.
     
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