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 Rn. (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 Rn, 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 Rn, 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.