Column space, null space and rank

Column space of matrix \(A\)

Span of matrix \(A\)’s column vectors.
Intuitively, it can also be seen as span of changed basis, because \(A\) changes the \(i'th\) basis into it’s \(i'th\) column.

Null space(kernel) of matrix \(A\)

Linear subspace of the domain of the map which is mapped to the zero vector.
As \(A\) can be seen as linear transformation, vector in \(A\)’s nullspace squeezes to zero when transformed by \(A\).

Rank of matrix \(A\)

The definition of rank is the dimension of vectorspace generated by columns of \(A\).
Intuitively, if \(A\) squeezes \(m\)-dimensional space into \(n\)-dimensional space, \(n\) is the rank of \(A\)