Let v 1 1 1 0 0 and w 1 0 0 1 1 be vectors in R5 Find the

Let v = (1, 1, 1, 0, 0) and w = (1, 0, 0, -1, -1) be vectors in R^5. Find the dimension and basis for the subspace S = {u element R^5: u v = 0 and u w = 0}

Solution

Let, u=(a,b,c,d,e)

u.v=0 and u.w=0 gives us two equations

a+b+c=0         ie      a=-b-c

a-d-e=0

We ahve five variables and two equations so 3 free variables

e=a-d=-b-c-d

So,

u=(-b-c,b,c,d,-b-c-d)=b(-1,1,0,0,-1)+c(-1,0,1,0,-1)+d(0,0,0,1,-1)

So basis for S is

{(-1,1,0,0,-1),(-1,0,1,0,-1),(0,0,0,1,-1)}