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3 Show that if A is an m×n matrix, then the solution set V to the equation Ax = 0 is a subspace of Rn Solution A1) Let x1;x2 ∈ Rn be two solutions to the equation Ax = 0 (that is, x1;x2 ∈ V)Then x1 x2 ∈ Rn, and A(x1 x2) = Ax1 Ax2 = 00 = 0 Thus x1 x2 ∈ V M1) Let x1 ∈ V, k ∈ R Then kx1 ∈ Rn, and A(kx1) = k(Ax1) = k(0) = 0 Thus kx1 ∈ V as well Thus by the subspaceA vector space V is a collection of objects with a (vector) addition and scalar multiplication defined that closed under both operations and which in addition satisfies the following axioms (i) (αβ)x = αxβx for all x ∈V and α,β∈F (ii) α(βx)=(αβ)x (iii) xy = y x for all x,y ∈V (iv) x(y z)=(xy)z for all x, y, z ∈VD u v Z l v } v o Z l v µ v d Z u u v µ Ç Á Z À } } v Z Á v µ X í ó X ì ì U ^ Z u î ì X ì ì

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