A vector space is a set of vectors that is closed under addition and scalar multiplication. In other words, given two vectors, v and w, a vector space is formed by the set of all linear combinations formed between v and w, namely cv + dw for arbitrary coefficients c and d.
Columns spaces and null spaces are special categories of vector spaces that have interesting properties related to systems of linear equations, Ax = b. The column space of the matrix A, C(A), is simply the linear combinations of A's column vectors. This implies that Ax = b may only be solved when b is a vector in A's column space. Finally, the null space of matrix A is another vector space formed by all the solutions to Ax = 0.