Discrete Mathematics -- Finding the basis of a given set

Question

Let A = {( 2z - w, z, w, z) | z, w ∈ R } ⊂ R⁴. Find a basis for A and show that it is a basis,

Answer 1

(2z - w, z, w, z) = z(2, 1, 0, 1) + w(-1, 0, 1, 0)

Any element of set A can be written as a linear combination of (2, 1, 0, 1) and (-1, 0, 1, 0),

So, the set {(2, 1, 0, 1), (-1, 0, 1, 0)} spans A. Also, (2, 1, 0, 1) and (-1, 0, 1, 0) are linearly independent since they are not multiples of each other. So, the set {(2, 1, 0, 1), (-1, 0, 1, 0)} is a basis for set A.