# Determining the rank of a matrix

1. Dec 28, 2016

### influx

1. The problem statement, all variables and given/known data

2. Relevant equations
N/A
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. Dec 28, 2016

### Staff: Mentor

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

3. Dec 28, 2016

### Ray Vickson

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.

4. Dec 28, 2016

### Ray Vickson

I hope you mean linear dependence.

5. Dec 28, 2016

### Staff: Mentor

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. Dec 28, 2016

### Ray Vickson

OK: I found the wording confusing.

7. Dec 28, 2016

### Staff: Mentor

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.