Find a set of vectors

Question

Find a set of vectors {u,v} in R⁴ that spans the solution set of the equations 

x-y+z-3w=0

x+2y-z+3w=0

 

u=?

v=?

Answer 1

There is some latitude in choosing x, y, z, and w to satisfy the equations and form the spanning vectors, u and v.

 

For x - y + z - 3w = 0, we may choose u = [1,1,3,1] for x, y, z, and w.

 

For x + 2y - z + 3w = 0, we may choose v = [1,1,6,1].

 

Since u and v are not multiples of each other, they are linearly independent and they span the plane in R⁴.