# Determining the rank of a matrix

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?

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fresh_42
Mentor
You should try to prove the linear independence of the four rows. Or simply draw them in a plane.

Ray Vickson
Homework Helper
Dearly Missed

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.

Delta2
Ray Vickson
Homework Helper
Dearly Missed
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.

fresh_42
Mentor
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.

Ray Vickson
Homework Helper
Dearly Missed
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.

fresh_42
Mentor
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.