Question

Problem statement: Prove the following: Theorem: Let n, r, s be positive integers, and let v1, . . . , vr E Rn and wi, . . . , w, є Rn. If wi є span {v1, . . . , vr} for each i = 1, . . . , s, then spanfVi, . .., v-) -spanfvi, . .., Vr, W,...,w,) Suggestiorn: To see how the proof should go, first try the case s - 1, r 2..]

Answer 1

Let n, r, s be positive integers and let v₁,…,v_r ∈ Rⁿ . Also, let w₁,…,w_s ∈ Rⁿ. If w_i ∈ span{ v₁,…,v_r }, then each w_i ( 1 ≤ i ≤ n) is a linear combination of v₁,…,v_r .

Apparently, span{ v₁,…,v_r } ⊆ span{ v₁,…,v_r, w₁,…,w_s }.

Further, an arbitrary vector u (say) in span{ v₁,…,v_r, w₁,…,w_s } is a linear combination of the vectors in span{ v₁,…,v_r }. Now, since each w_i ( 1 ≤ i ≤ n) is a linear combination of v₁,…,v_r, hence u is a linear combination of the vectors v₁,…,v_r. Therefore u ∈ span{ v₁,…,v_r }.

Hence span{ v₁,…,v_r, w₁,…,w_s } ⊆ span{ v₁,…,v_r }.

Therefore, span{ v₁,…,v_r, w₁,…,w_s } = span{ v₁,…,v_r }.