Q:

What is the form of bases in a non square matrix?

I am trying to find the null space and the column space for this matrix.
$\pmatrix{1&1&1&1\\1&2&0&3\\0&1&-1&2}$
The process I usually follow is to first convert to REF, which gives:
$\pmatrix{1&0&2&-1\\0&1&-1&2\\0&0&0&0}$
Which, reading from REF, seems to give a basis of the form $\pmatrix{-2\\1\\1\\0}\pmatrix{1\\-2\\0\\1}$ for the null space. It looks a little odd, but I suppose it makes sense that a matrix with $n=4$ would have vectors of $4$ entries.
Where I am really confused is how to find a basis for col(A). I would usually just take the two pivot columns and call that a basis, but clearly those vectors would not be the same size as the ones in null(A). How else would I find col(A) then?

A:

In general, the null space and column space of a matrix don't have to be the same. By definition, the column space of a matrix is the span of its columns; since your matrix here has 3 rows, all of its columns are vectors in $\mathbb{R}^3$. As such, its basis should not look like your null space basis. The same procedure you describe, taking the pivot columns, suffices to give you a basis.

