Dimension of an affine subspace

The set in {mbox{bf R}}^3 defined by the linear equations
 x_1 - 13 x_2 + 4 x_3 = 2, ;; 3 x_2 - x_3 = 9
is an affine subspace of dimension 1. The corresponding linear subspace is defined by the linear equations obtained from the above by setting the constant terms to zero:
 x_1 - 13 x_2 + 4 x_3 = 0, ;; 3 x_2 - x_3 = 0
We can solve for x_3 and get x_1 = x_2, x_3 = 3x_2. We obtain a representation of the linear subspace as the set of vectors x in {mbox{bf R}}^3 such that
 x = left(begin{array}{c} 1  1  3 end{array}right) x_2.
Hence the linear subspace is the span of the vector (1,1,3), and is of dimension 1.