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

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=(a,b,c,d) be in W\' (We denote Orthogonal complement of W by W\')

Taking dot product with each of the three vectors gives

a-2c=0 ie a=2c

2b+4c-d=0 ie d=2b+4c

2a+2b=d ( This is equivalent to first two equations)

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

So basis for W\' is

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