显然
\begin{equation*}
\alpha_i=0\alpha_1+\cdots+1\alpha_i+\cdots+0\alpha_n\in\langle\alpha_1,\ldots,\alpha_n\rangle,
\end{equation*}
所以\(\langle\alpha_1,\ldots,\alpha_n\rangle\)是\(\F^m\)的非空子集。对任意\(\alpha,\beta\in\langle\alpha_1,\ldots,\alpha_n\rangle,c\in\F\),存在\(a_1,\ldots,a_n,b_1,\ldots,b_n\in\F\),使得
\begin{equation*}
\alpha=a_1\alpha_1+\cdots+a_n\alpha_n,\ \beta=b_1\alpha_1+\cdots+b_n\alpha_n,
\end{equation*}
利用加法交换律、结合律及加法对数乘的分配律可知
\begin{equation*}
\alpha+\beta=(a_1+b_1)\alpha_1+\cdots+(a_n+b_n)\alpha_n\in\langle\alpha_1,\ldots,\alpha_n\rangle,
\end{equation*}
\begin{equation*}
c\alpha=(ca_1)\alpha_1+\cdots (ca_n)\alpha_n\in\langle\alpha_1,\ldots,\alpha_n\rangle,
\end{equation*}
因此\(\langle\alpha_1,\ldots,\alpha_n\rangle\)是\(\F^m\)中包含\(\alpha_1,\ldots,\alpha_n\)的子空间。
若\(V\)是\(\F^m\)中包含\(\alpha_1,\ldots,\alpha_n\)的子空间,由子空间定义可知,对任意\(c_1,\ldots,c_n\in\F\),
\begin{equation*}
c_1\alpha_1+\cdots+c_n\alpha_n\in V,
\end{equation*}
故\(\langle\alpha_1,\ldots,\alpha_n\rangle\subseteq V\),因此\(\langle\alpha_1,\ldots,\alpha_n\rangle\)是\(\F^m\)中包含\(\alpha_1,\ldots,\alpha_n\)的最小子空间。