Question

Find a basis for the orthogonal complement of the subset U = span{ (1,1,1,0,1), (1,?1,0,1,0), (?1,0,1,0,1) } in R5.

please answer clearly

Answer 1

Given, U = Span{u,v,w}, where u = (1,1,1,0,1), v = (1,-1,0,1,0) and w = (-1,0,1,0,1).

Let x = (a,b,c,d,e) be the orthogonal complement of the subset U.

Then, (x,u) = 0, (x,v) = 0, (x,w) = 0

i.e., a+b+c+e = 0

a-b+d = 0

-a+c+e = 0

i.e., a = c+e

b = -2c-2e

d = -3c-3e

Let us take c = m and e = n. Then, a = m+n, b = -2m-2n, d = -3m-3n.

Therefore, (a,b,c,d,e) = (m+n,-2m-2n,m,-3m-3n,n) = m(1,-2,1,-3,0)+n(1,-2,0,-3,1)

Hence, the required basis for the orthogonal complement of the subset U is {(1,-2,1,-3,0),(1,-2,0,-3,1)}.