I've seen theoretical examples that differentiate between the definitions of each and believe I understand the two concepts in theory, but I'm not sure how these actually appear differently when using Gaussian / Gauss-Jordan elimination. I've been told that if it is successfully placed in RREF and has rank equal to the number of rows in the original representation, then it is a spanning set, but wouldn't this also mean that it is linearly independent? I'm aware that the concepts are not mutually inclusive, but the aforementioned test seems to imply they are.

In short, an explanation of how to separately test for each using Gaussian / Gauss-Jordan elimination, where the test is not inclusive of both, would likely provide sufficient clarification for me.

Appreciate it!

Edit: I believe I have arrived at an answer. Would it be correct to say that the RREF matrix represents a spanning set iff rank = the number of rows, and is linearly independent iff rank = the number of columns? Apologies for the premature question if my new understanding is correct!

Follow-Up Edit: It appears as though I've been unclear. I'll provide a concrete example below.

Take this vector set:

[Vector set](https://i.stack.imgur.com/Z9pBR.png)

This vector set can be represented as the following matrix:

1 -1 -3

1 1 5

2 1 6

-1 0 -1

Placing it into RREF, I get

1 0 1

0 1 4

0 0 0

0 0 0

What I'm unclear on is how this can be used to determine whether or not the vector set is a spanning set, and whether or not it is linearly independent.

My new understanding is that if the rank (as easily determined by the new RREF) is = to the number of rows in the original representation (i.e there are no zero rows in RREF) then it is a spanning set, and if the rank is = to the number of columns in the original representation then it is linearly independent.

(So by my new understanding, it can be stated that it is neither a spanning set nor linearly independent, because the rank = 2, which is less than the number of rows (4, for the spanning set test) and less than the number of columns (3, for the linear independence test)

Is this correct, completely off, or not nuanced enough?