Which vector must be removed in a dependent set of vectors Ok. I have a big problem with this following example and it makes me really confused: 
We have here a set of vectors which is : vect{(1,2), (-1,0), (0,1)}
This set is dependent we have : 2*(0,1)-(-1,0)= (1,2)
I want to make it  linearly independent, so I have to remove one vector, the teacher in class said we can't remove (1,2), I forgot the reason why, but if we take any 2 vectors in that set and we check if it's linearly independent, we will find that it is linearly independent. It's too late for me to contact my teacher so I'm asking you people. Please  help! Pretend I am the dumbest student you ever met.
 A: You're right — that set is dependent, and removing any vector from it makes it independent. There's no mathematical reason why you can't remove $\langle 1, 2\rangle$, so perhaps there's something else going on.
It may be that removing $\langle 1, 2\rangle$ is the "easiest" case to prove and so your teacher wants you to have more practice proving one of the other cases.
After all, whenever 
$a\langle -1,0\rangle + b\langle0,1\rangle = \langle0,0\rangle$, then $\langle -a, b\rangle = \langle 0,0\rangle$, which shows that $a=b=0$, and proves that the two vectors are independent. That's a pretty straightforward proof.
Maybe it takes slightly more work to see that whenever
$a\langle 1,2\rangle + b\langle0,1\rangle = \langle0,0\rangle$, then $\langle a, 2a+b\rangle = \langle 0, 0\rangle$ Looking at the first component, we notice that $a=0$. This means that $\langle 0, 0+b\rangle = \langle 0,0\rangle$, which proves that also $b=0$. This proves that the two vectors are independent.
A: I guess that your teacher have in mind that the basis is an "ordered basis". That's is some important for the use of the coordinates of some vector expressed in that basis, change basis matrix, etc. For ordered basis, the general rule for discard vectors is using the row echelon form (reduced or not) of the matrix formed with the vector coordinates. If they are as rows, then the non-zero rows correspond to the independent vectors, and if the arrange is as columns of the matrix, the independent vectors are those which have pivot values (i.e. nonzero numbers in the diagonal).
