Question

Let L be a homogeneous linear system involving m equations and n real variables. Let H be the solution set of L. Prove that H is a subspace of R^n.

Answer 1

Let L be a homogeneous linear system involving m equations and n real variables and let H be the solution set of L. Let A be the coefficient matrix of this linear system of equations.

Further, let X = (a₁,a₂,…,a_n) and Y = (b₁,b₂,…,b_n) be 2 elements of H. Also, let k be an arbitrary real scalar. Then AX = 0 and AY = 0 so that A(X+Y) = AX+AY = 0+0 = 0. This implies that H is closed under vector addition. Also, A(kX) = kAX = k.0 = 0 so that H is closed under scalar multiplication.

Further, the zero vector (0,0,…,0) is apparently in H as zero is always the solution of a homogeneous linear system of equations. Therefore, H is a vector space, and since H is a subset of Rⁿ, hence H is a subspace of Rⁿ.