# Different basis for the column space of same matrix.

So i am a bit confused by the column space of a matrix. I seem to read various answers in different places as to what actually spans the column space of a matrix.

1) Lets say i have a matrix, most places seems to suggest that one should find the reduced row echelon form of the matrix and pick the columns with leading 1's from the original matrix.

2) Other places i've seen seem to suggest that the columns with leading 1's from the RREF matrix also makes up the column space.

Are both of these ways correct? and if so why is 2) correct?

• 2) is incorrect. It's only telling you which columns to pick from the original matrix. (When you do row operations, you mess with the entries enough that the columns of the RREF have nothing to do with the original columns. Only the rows can be recovered by linear combinations.) – Ted Shifrin May 20 '18 at 0:26
• @TedShifrin Thank you. That was the answer i was looking for. – sn3jd3r May 20 '18 at 17:04

So if you want to find a basis of the column space of a matrix $A$, then 1) is the correct method: You use an echelon form of $A$ to determine which columns of $A$ you should take.