首先，我们证明存在性。由于\((\xi_{1}, \dots, \xi_{n})\)是\(V\)的基，\(V\)中任一向量\(\alpha \in V\)可被基向量唯一表出
\begin{equation*}
\alpha = a_{1} \xi_{1} + \dots + a_{n} \xi_{n}, \quad a_{1},\dots,a_{n} \in \F.
\end{equation*}
考虑\(V\)到\(U\)的映射\(\varphi\)：
\begin{equation*}
\varphi(\alpha) = a_{1} \beta_{1} + \dots + a_{n} \beta_{n}, \quad \forall \alpha \in V.
\end{equation*}
则显然\(\varphi(\xi_{i}) = \beta_{i}, \forall i \in [n]\)。我们验证\(\varphi\)确实是线性映射。 对于任意\(\alpha = \sum_{i=1}^{n} a_{i} \xi_{i}, \beta = \sum_{i=1}^{n} b_{i} \xi_{i} \in V\)，有
\begin{equation*}
\varphi(\alpha + \beta) = \varphi\bigg( \sum_{i=1}^{n} (a_{i} + b_{i}) \xi_{i} \bigg) = \sum_{i=1}^{n} (a_{i} + b_{i}) \beta_{i}.
\end{equation*}
进一步展开，并根据\(\varphi\)的定义有
\begin{equation*}
\varphi(\alpha + \beta) = \sum_{i=1}^{n} a_{i} \beta_{i} + \sum_{i=1}^{n} b_{i} \beta_{i} = \varphi(\alpha) + \varphi(\beta).
\end{equation*}
所以\(\varphi\)保持加法运算。类似地，对于任意\(\alpha = \sum_{i=1}^{n} a_{i} \xi_{i}\)以及\(c \in \F\)有
\begin{equation*}
\varphi(c \alpha) = \sum_{i=1}^{n} c a_{i} \beta_{i} = c \sum_{i=1}^{n} a_{i} \beta_{i} = c \varphi(\alpha).
\end{equation*}
所以\(\varphi\)保持数乘运算。综上，\(\varphi\)确实为线性映射，存在性成立。
下面证明唯一性。假设存在其他线性映射\(\psi \in \mathcal{L}(V,U)\)使得\(\psi(\xi_{i}) = \beta_{i}, \forall i = 1, \dots, n\)。 由于\(\psi\)是线性映射，而任意的向量\(\alpha \in V\)可被线性表出为\(\alpha = a_{1} \xi_{1} + \dots + a_{n} \xi_{n}\)，所以有
\begin{equation*}
\psi(\alpha) = a_{1} \psi(\xi_{1}) + \dots + a_{n} \psi(\xi_{n}) = a_{1} \beta_{1} + \dots + a_{n} \beta_{n}.
\end{equation*}
又由\(\varphi\)的定义，我们有
\begin{equation*}
\psi(\alpha) = a_{1} \varphi(\xi_{1}) + \dots + a_{n} \varphi(\xi_{n}) = \varphi(\alpha).
\end{equation*}
因此，\(\psi\)与\(\varphi\)是同一个映射。
