# Determining the rank of a matrix

## Homework Statement 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

## Homework Statement 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.