Suppose that W span1 0 2 0 0 2 4 1 2 2 0 1 Find a basis for

Suppose that W = span({1, 0, -2, 0), (0, 2, 4, -1), (2, 2, 0, -1)}). Find a basis for the orthogonal complement of W

Solution

Let, x be in orthogonal complement of W

x=(a,b,c,d)

x.(1,0,-2,0)=0

a=2c

x.(0,2,4,-1)=0
2b+4c-d=0

d=2b+4c

x.(2,2,0,-1)=d

d=2a+2b=2b+4c

So,

x=(2c,b,c,2b+4c)=b(0,1,0,2)+c(2,0,1,4)

HEnce basis is

{(0,1,0,2),(2,0,1,4)}