Statement:

  Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{ S} \) be a non empty finite subset of \(\pmb{ V} \). Then the set \(\pmb{L(S) } \) is the smallest subspace of \(\pmb{V} \) containing \(\pmb{ S} \).

  Proof:
  Given that \(\pmb{(V,+,\cdot)} \) is a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{ S} \) is a non empty finite subset of \(\pmb{ V} \).
  To prove \(\pmb{L(S) } \) is the smallest subspace of \(\pmb{V} \) containing \(\pmb{ S} \).
  Let \(\pmb{ S=\{\alpha_{1},\alpha_{2},…,\alpha_{n} \} } \).

1. To prove \(\pmb{L(S)\ne \Phi} \)
   Since \(\pmb{ 0\in F} \) then \(\pmb{\theta=\displaystyle\sum_{i=1}^{n} 0\cdot\alpha_{i}\in L(S)} \). Therefore \(\pmb{L(S)\ne \Phi} \).

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2. To prove \(\pmb{\beta+\gamma\in L(S)~\forall~\beta,\gamma\in L(S)} \)
   Let \(\pmb{\beta,\gamma\in L(S)} \)
   Then \(\pmb{\exists} \) scalars \(\pmb{c_{i},d_{i}\in F} \) where \(\pmb{i=1,2,…,n} \) such that
   \(\pmb{\beta=\displaystyle\sum_{i=1}^{n} c_{i}\cdot\alpha_{i}} \) and \(\pmb{\gamma=\displaystyle\sum_{i=1}^{n} d_{i}\cdot\alpha_{i}} \)
   Now \(\pmb{\beta+\gamma=\displaystyle\sum_{i=1}^{n} c_{i}\cdot\alpha_{i}+\displaystyle\sum_{i=1}^{n} d_{i}\cdot\alpha_{i}} \)
   \(\implies\pmb{\beta+\gamma=\displaystyle\sum_{i=1}^{n}(c_{i}\cdot\alpha_{i}+d_{i}\cdot\alpha_{i} ) } \)
   \(\implies\pmb{\beta+\gamma=\displaystyle\sum_{i=1}^{n}(c_{i}+d_{i} )\cdot\alpha_{i} } \)
   \(\implies\pmb{\beta+\gamma\in L(S) } \) since \(\pmb{c_{i}+d_{i}\in F} \) where \(\pmb{i=1,2,…,n} \).

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3. To prove \(\pmb{k\cdot\delta\in L(S)~\forall~\delta\in L(S), k\in F} \)
   Let \(\pmb{\delta\in L(S)} \) and \(\pmb{k\in F} \)
   Then \(\pmb{\exists} \) scalars \(\pmb{q_{i}\in F} \) where \(\pmb{i=1,2,…,n} \) such that
   \(\pmb{\delta=\displaystyle\sum_{i=1}^{n} q_{i}\cdot\alpha_{i}} \)
   Now \(\pmb{k\cdot\delta=k\cdot\left(\displaystyle\sum_{i=1}^{n} q_{i}\cdot\alpha_{i} \right) } \)
   \(\implies \pmb{k\cdot\delta=\displaystyle\sum_{i=1}^{n} k\cdot\left(q_{i}\cdot\alpha_{i} \right) } \)
   \(\implies \pmb{k\cdot\delta=\displaystyle\sum_{i=1}^{n} \left(k\cdot q_{i}\right)\cdot\alpha_{i} } \)
   \(\implies\pmb{k\cdot\delta\in L(S) } \) since \(\pmb{k\cdot q_{i}\in F} \) where \(\pmb{i=1,2,…,n} \).

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4. To prove \(\pmb{L(S) } \) is the smallest subspace of \(\pmb{V} \) containing \(\pmb{ S} \)
   Let \(\pmb{K } \) be any subspace of \(\pmb{V} \) containing \(\pmb{ S} \). That is \(\pmb{ S\subseteq K} \).
   To prove \(\pmb{L(S)\subseteq K } \)
   Let \(\pmb{\eta\in L(S)} \)
   Then \(\pmb{\exists} \) scalars \(\pmb{r_{i}\in F} \) where \(\pmb{i=1,2,…,n} \) such that
   \(\pmb{\eta=\displaystyle\sum_{i=1}^{n} r_{i}\cdot\alpha_{i}} \)
   Since \(\pmb{r_{i}\in F} \) and \(\pmb{\alpha_{i}\in K} \), \(\pmb{i=1,2,…,n} \) then
   \(\pmb{r_{i}\cdot \alpha_{i}\in K }\), \(\pmb{i=1,2,…,n} \) since \(\pmb{K } \) is a subspace of \(\pmb{V} \).
   \(\implies \pmb{\displaystyle\sum_{i=1}^{n} r_{i}\cdot \alpha_{i}\in K }\) since \(\pmb{K } \) is a subspace of \(\pmb{V} \).
   \(\implies \pmb{\eta\in K }\)
   Therefore \(\pmb{L(S)\subseteq K } \).
   Since \(\pmb{K } \) is any subspace of \(\pmb{V} \) containing \(\pmb{ S} \), hence \(\pmb{L(S) } \) is the smallest subspace of \(\pmb{V} \) containing \(\pmb{ S} \).