You can consider the columns of a matrix as a **set** of (column) vectors. The space they span is the column space.
For example the matrix
1 2
3 4
5 6
has two column vectors, (1 3 5) and (2 4 6), which span a two-dimensional subspace of **R**^3
The null space is just the set of vectors going into 0 when you apply the matrix on them. If A is the matrix and x the vector, the null space consists of all vectors solution of the equation
A.x = 0
It’s easy to show that they form a subspace (of the space where you take your x-es).
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