Solution set of a linear equation

Theorem

The solution set of a linear equation
 mathbf{A} = left{ x in mathbf{R}^n ~:~ Ax = y right},
where A in mathbf{R}^{m times n} and y in mathbf{R}^m are given, is either empty, of an affine set.

Proof: Indeed, if it is not empty, we can express the condition Ax=y as A(x-x_0) = 0, where x_0 is a particular solution. Hence, x in mathbf{A} if and only if x-x_0 in mathbf{L}, where mathbf{L} = { x ::: Ax = 0} is a subspace. We write this as mathbf{A} = x_0+mathbf{L}. Thus, mathbf{A} is an affine subspace.