What Determines Pivot Positions in a Matrix?

[NorwegianStud]

Matrix: Ã=
1 -2 0 0 3
2 -5 -3 -2 6
0 5 15 10 0
2 6 18 8 6

Matrix on reduced echelon form:
1 0 0 -2 3
0 1 0 -1 0
0 0 1 1 0
0 0 0 0 0

X 0 0 -2 3
0 X 0 -1 0
0 0 X 1 0
0 0 0 0 0

Are the pivot positions the ones I've marked with X? And therefor colum 1, 2 and 3 are pivot columns? Or have I completely misunderstood?
Bonus question: Can't work out "Find a basis for Col(Ã), Row(Ã)". What does that even mean?

 

[QuarkCharmer]

Those are indeed the pivots.

As for the bonus, the basis is just a set of linearly independent vectors that can represent every vector in a space.

 

[NorwegianStud]

Those are indeed the pivots.

As for the bonus, the basis is just a set of linearly independent vectors that can represent every vector in a space.

Great, thanks a lot.

Linearly independent vectors? I've seen that before when I looked for the answer, but I didn't quite understand it. How do I check if it's linearly independent?