Find a basis for the subspace of all vectors x1 x2 x3 x4 in

Find a basis for the subspace of all vectors (x1, x2, x3, x4) in R 4 such that x1 + x2 = x3 + x4. The following two questions are about subsets of the set of real-valued functions of the real line. We will call this set F.

Solution

4 variable and 1 euqation so there will be 3 free variables

x1+x2=x3+x4

x1=-x2+x3+x4

So general vectore in R4 is

(-x2+x3+x4,x2,x3,x4)=x2(-1,1,0.0)+x3(1,0,1,0)+x4(1,0,0,1)

HEnce basis for subspace is

{(-1,1,0.0),(1,0,1,0),(1,0,0,1)}