Answer to Question #258815 in Linear Algebra for Bere

Question

Let W be the set of vectors in R³ of the form(a,b,½a-2b)

show that W is subspace of R³

Answer 1

Let m, n ∈ W and Let β∈ℝ. Then m = (a, b, ½a-2b) and n = (x, y, ½x-2y)

We shall show that m + βn ∈ W

m + βn = (a, b, ½a-2b) + β(x, y, ½x-2y)

= (a, b, ½a-2b) + (βx, βy, β(½x-2y))

= (a+βx, b+βy, ½a-2b + β(½x-2y))

Since a+βx, b+βy, ½a-2b + β(½x-2y) ∈ ℝ ∀ a, b, x, y, β ∈ ℝ, then m+βn ∈ ℝ.

Thus, W is a subspace of ℝ³