The null space of an invertible 3x3 matrix is the 0 matrix, because invertibility implies that applying a finite sequence of elementary row operations to a matrix eventually yields the identity matrix. What is the solution space to the homogeneous system Ax = 0, where A is any invertible matrix? Since A is row and column equivalent to the identity matrix, isn't it the zero vector? Also, rank-nullity theorem: rank + nullity = n, where n is the number of columns. If the matrix is invertible, then rank is equal to the number of columns and nullity = 0.
As for the column space, it is simply the span of all of the columns of an invertible matrix. Why? Because the rank for an invertible matrix is equal to the number of columns by the rank-nullity theorem. Intuitively, none of the columns disappear (linearly dependence) for an invertible matrix; all of the columns are linearly independent and even form a basis for the column space.
This can be generalized to the row space as well.