If W is a subspace of R3 where W span1 7 2 0 8 0 0 1 1 6 0

If W is a subspace of R^3 where W = span{(1, 7, -2, 0, -8), (0, 0, 1, 1, 6), (0, 0, 0, 1, 3)}, find a basis for W

Solution

We denote W^perpendicular by W\'

Let (a,b,c,d,e) be in W\'

So we need that

(a,b,c,d,e).(1,7,-2,0,-8)=0

a+7b-2c-8e=0

(a,b,c,d,e).(0,0,1,1,6)=0

c+d+6e=0

(a,b,c,d,e).(0,0,0,1,3)=0

d+3e=0

d=-3e

c=-d-6e=3e-6e=-3e

c=-3e

a+7b-2c-8e=0

a+7b+6e-8e=0

a=-7b+2e

So,

(a,b,c,d,e)=(-7b+2e,b,-3e,-3e,e)=b(-7,1,0,0,0)+e(2,0,-3,-3,1)

So basis is

{(-7,1,0,0,0),(2,0,-3,-3,1)}