Dimension of an affine subspace

The set {bf L} 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 . 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 $x in {mbox{bf R}}^3$ that have the form

x = left(begin{array}{c} 1 1 3 end{array}right) t,

for some scalar t=x_2 . Hence the linear subspace is the span of the vector u :=(1,1,3) , and is of dimension .

We obtain a representation of the original affine set by finding a particular solution x^0 , by setting say x_2=0 and solving for x_1,x_3 . We obtain

$x^0 = left(begin{array}{c} 38 0 -9 end{array}right).$

The affine subspace {bf L} is thus the line $x^0+mbox{bf span}(u)$ , where x^0,u are defined above.