Find a set of vectors {𝑢⃗ ,𝑣⃗ } in ℝ^4 that spans the solution set of the equations: w-x+2y-z=0 4w+2x-y+2z=0

Question

Find a set of vectors {𝑢⃗ ,𝑣⃗ } in ℝ^4 that spans the solution set of the equations: w-x+2y-z=0 4w+2x-y+2z=0

Answer 1

The rref is

w +1/2y = 0 and x -3/2y+z = 0. Let y = r and z = s

w = -1/2 r + 0s

x = 3/2 r - s

y = r + 0 s

z = 0 r + s

Hence <w, x, y, z> is generated by the two vectors

u = <-1/2, 3/2, 1, 0>

v = <0, -1, 0, 1>