Represent the set {(a − b, b − c, c − a) | a, b, c ∈ R} ⊂ R 3

Question

Represent the set {(a − b, b − c, c − a) | a, b, c ∈ R} ⊂ R 3 as Span{v₁, v₂, v₃} for some vectors v₁, v₂, v₃. Prove that this set is a subspace of R³.

Answer 1

Since a, b, and c behave as free variables, we have:

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

which you can verify directly. Hence the given set is equal to Span(v1, v2, v3) where:

1. v1= (1,0,-1)
2. v2 = (-1,1,0)
3. v3 = (0,1,-1)

The Span of any set of vectors is always a subspace; the proof uses the standard subspace test (show that Span contains the zero vector and is closed under vector addition and scalar multiplication). The notation is a bit inconvenient, but you can find the general proof in most intro Linear Algebra books.