Determining the rank of a matrix

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Homework Statement


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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?
 

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  • #2
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You should try to prove the linear independence of the four rows. Or simply draw them in a plane.
 
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Ray Vickson
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Homework Statement


2136c8.png


Homework Equations


N/A

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?
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.
 
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Ray Vickson
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You should try to prove the linear independence of the four rows. Or simply draw them in a plane.
I hope you mean linear dependence.
 
  • #5
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I hope you mean linear dependence.
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.
 
  • #6
Ray Vickson
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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.
OK: I found the wording confusing.
 
  • #7
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OK: I found the wording confusing.
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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