Prove that the set {v + w, v + x, w − x} is always linearly dependent

Question

Let v, w, x ∈ Rⁿ be vectors.

Prove that the set {v + w, v + x, w − x} is always linearly dependent (even if {v, w, x} is linearly independent).

Answer 1

Let:

a = v + w = [ a₁ a₂ ... a_n ]

b = v + x = [ b₁ b₂ ... b_n ]

c = w - x = [c₁ c₂ ... c_n ]

and note that a₁ = v₁+w_1, b₁ = v₁+x₁, c₁ = w₁-x₁, etc. ...

Then

a₁ - b₁ = (v₁+w₁)-(v₁+x₁) = (w₁-x₁) = c₁

a₂ - b₂ = (v₂+w₂)-(v₂+x₂) = (w₂-x₂) = c₂

...

a_n - b_n = (v_n+w_n)-(v_n+x_n) = (w_n-x_n) = c_n

This shows that the three vectors {a,b,c} are linearly dependent.

Answer 2

1*(v+w) - 1*(v+x) - 1*(w-x) = 0 [when you expand, each of w, v, x appears once with a coefficient of +1 and once with a coefficient of -1]

This is a linear combination of v+w, v+x, w-x in which the coefficients are not all zero. Therefore the vectors are linearly dependent.