Solution to Selected Homework Problems
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چکیده
21. Proof: For any x ∈ Span(v + cw,w), there exists scalars a, b such that x = a(v+cw)+bw. Rewriting x = av+(ac+b)w, we have x ∈ Span(v,w). Therefore, Span(v + cw,w) ⊂ Span(v,w). On the other hand, for any x ∈ Span(v,w), there exists scalars a, b such that x = av+bw. We need to rewrite x = a′(v+cw)+b′w for some scalars a′, b′, in order to show that x ∈ Span(v+cw,w). Clearly, a′ = a, b′ = b−ac satisfy the equation. Therefore, we conclude that Span(v,w) ⊂ Span(v+ cw,w).
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