In this calculator, the solution sets of homogeneous linear systems provide a really important source of vector spaces. So, let A be a m cross n matrix of any order, also consider the homogeneous system.

Since A is m by n, the set of all vectors ‘x’ that satisfy this equation forms a subset of R^{n}. (Note that this subset is non-empty obviously, since it clearly contains the zero vector: x = 0 always satisfies A x = 0).

Now this subset actually forms a subspace of R^{n}, which is called the ‘nullspace’ of the matrix A and denoted by N(A). In order to prove that N(A) is a subspace of R^{n}, the closure under both addition and scalar multiplication must be established.

Again, if x _{1} and x _{2} are in N(A), then, by definition, A x _{1} = 0 and A x _{2} = 0. Adding these equations yields that verifies the closure under addition.

Next, if x is in N(A), then A x = 0, so if k is any scalar, hence verifying closure under scalar multiplication. Thus, the solution set of a fairly homogeneous linear system forms a vector space.

Note carefully that if the system is not particularly homogeneous, then the set of solutions is also not a vector space since the set will not contain the zero vector.