Assume $\{v_1,…,v_k\}$ spans a subset $S$ of $\mathbb{R}^n$ and $\{w_1,…,w_ℓ\}$ spans a subset $T$ of $\mathbb{R}^n$.

Question

Assume $\{v_1,…,v_k\}$ spans a subset $S$ of $\mathbb{R}^n$ and $\{w_1,…,w_l\}$ spans a subset $T$ of $\mathbb{R}^n$. Define $S+T=\{s+t ∣s\in S,t\in T\}.$ Prove that Span$\{v_1,…,v_k,w_1,…,w_l\}=S+T.$

I've struck by the meaning of $s+t.$ I need a completed answer.

Answer 1

It's clear that span$\{v_1,…,v_k,w_1,…,w_l\}\subset S+T.$ For the converse, for any element $s+t$ in $S+T,$ $s$ and $t$ can be expressed in $S$'s and $T$'s bases respectively. Then the conclusion follows.