Basis in $mbox{bfLarge R}^3$

The set of three vectors in ${mathbf{R}}^3$ :

$x_1 := left(begin{array}{c} 1 2 3 end{array}right), ;; x_2 = left(begin{array}{c} 4 5 6 end{array}right), ;; x_3 = left(begin{array}{c} 3 3 3 end{array}right),$

is not independent, since x_1 -x_2 + x_3 = 0 , and its span has dimension . Since x_1,x_2 are independent (the equation lambda_1 x_1 + lambda_2 x_2 = 0 has lambda = 0 as the unique solution), a basis for that span is, for example, ${x_1,x_2}$ . In contrast, the collection ${x_1,x_2,x_3-e_1}$ spans the whole space ${mathbf{R}}^3$ , and thus forms a basis of that space.