Let W be the subspace of all vectors in R4 whose components

Let W be the subspace of all vectors in R⁴ whose components add up to 0. Find a basis for W. (You must show that it is a basis. See the definition of a basis.)

Solution

Vector in W is: (r,s,u,v)

r+s+u+v=0

r=-s-u-v

So vector becomes: (-s-u-v,s,u,v)=s(-1,1,0,0)+u(-1,0,1,0)+v(-1,0,0,1)

HEnce basis for W is

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

Clearly it spans W because we have taken a general vector in W above and shown that it lies in span of these three vectors. IF we can show these three vectors are linearly independent then the proof is done that thye form basis for W

Let, a,b,c so that

a(-1,1,0,0)+b(-1,0,1,0)+c(-1,0,0,1)=0

So,

a+b+c=0

a=0,b=0,c=0

Hence the vectors are linearly independent

Hence it forms a basis for W