必要性：若
\(V\)与
\(U\)同构，则存在从
\(V\)到
\(U\)的同构映射
\(\varphi\)，由命题
6.2.16可知，
\(\varphi\)将
\(V\)的一组基映射成
\(U\)的一组基，因此
\(V\)与
\(U\)维数相同。
充分性：设\(\dim V = \dim U = n\)，令\((\alpha_{1}, \ldots, \alpha_{n}) \in V\)为\(V\)的基，\((\beta_{1}, \ldots, \beta_{n}) \in U\)为\(U\)的基。由于\((\alpha_{1}, \ldots, \alpha_{n})\)是\(V\)的基，任意\(\alpha \in V\)可由改组基唯一地线性表示为\(\alpha = c_{1} \alpha_{1} + \cdots + c_{n} \alpha_{n}\)，其中\(c_{1}, \ldots, c_{n} \in \F\)。我们定义\(V\)到\(U\)的映射\(\varphi\)如下：
\begin{equation*}
\varphi(\alpha) := c_{1} \beta_{1} + \cdots + c_{n} \beta_{n}, \quad \forall \alpha = c_{1} \alpha_{1} + \cdots + c_{n} \alpha_{n} \in V.
\end{equation*}
下面证明\(\varphi\)是\(V\)到\(U\)的同构映射。首先说明\(\varphi\)是单射。对于任意\(\gamma_{1}, \gamma_{2} \in V\)，设
\begin{equation*}
\gamma_{1} = a_{1} \alpha_{1} + \cdots + a_{n} \alpha_{n}, \quad \gamma_{2} = b_{1} \alpha_{1} + \cdots + b_{n} \alpha_{n}, \quad a_{i}, b_{j} \in \F, i,j=1,\ldots, n.
\end{equation*}
若\(\varphi(\gamma_{1}) = \varphi(\gamma_{2})\)，由\(\varphi\)的定义有
\begin{equation*}
a_{1} \beta_{1} + \cdots + a_{n} \beta_{n} = b_{1} \beta_{1} + \cdots + b_{n} \beta_{n} \quad \iff \quad (a_{1} - b_{1}) \beta_{1} + \cdots + (a_{n} - b_{n}) \beta_{n} = 0.
\end{equation*}
又因为\(\beta_{1}, \ldots, \beta_{n}\)线性无关，所以\(a_{i} = b_{i}, \forall i=1,\ldots,n\)，即\(\gamma_{1}=\gamma_{2}\)。因此，\(\varphi\)是单射。
接下来，说明\(\varphi\)是满射。设\(\beta\)为\(U\)中任一向量，由于\((\beta_{1}, \ldots, \beta_{n})\)是\(U\)的基，存在唯一一组\(d_{1}, \ldots, d_{n} \in \F\)使得
\begin{equation*}
\beta = d_{1} \beta_{1} + \cdots + d_{n} \beta_{n}.
\end{equation*}
我们令\(\alpha := d_{1} \alpha_{1} + \cdots + d_{n} \alpha_{n} \in V\)，由\(\varphi\)的定义可知\(\varphi(\alpha) = \beta\)，即\(\beta\)可在\(V\)中找到\(\varphi\)下的原像。由\(\beta\)的任意性，\(\varphi\)是满射。
最后，我们说明\(\varphi\)保持线性运算。对于任意\(\alpha_{1}, \alpha_{2} \in V\)，设
\begin{equation*}
\alpha_{1} = a_{1} \alpha_{1} + \cdots + a_{n} \alpha_{n}, \quad \alpha_{2} = a'_{1} \alpha_{1} + \cdots + a'_{n} \alpha_{n}.
\end{equation*}
考虑任意\(c_{1}, c_{2} \in \F\)，根据\(\varphi\)的定义，我们有
\begin{equation*}
\varphi(c_{1} \alpha_{1} + c_{2} \alpha_{2}) = \varphi( (c_{1} a_{1} + c_{2} a'_{1}) \alpha_{1} + \cdots + (c_{1} a_{n} + c_{2} a'_{n}) \alpha_{n} ) = (c_{1} a_{1} + c_{2} a'_{1}) \beta_{1} + \cdots + (c_{1} a_{n} + c_{2} a'_{n}) \beta_{n}.
\end{equation*}
结合\(U\)中加法和数乘的性质以及\(\varphi\)的定义可得，
\begin{equation*}
\varphi(c_{1} \alpha_{1} + c_{2} \alpha_{2}) = c_{1} (a_{1} \beta_{1} + \cdots + a_{n} \beta_{n}) + c_{2} (a'_{1} \beta_{1} + \cdots + a'_{n} \beta_{n}) = c_{1} \varphi(\alpha_{1}) + c_{2} \varphi(\alpha_{2}).
\end{equation*}
根据命题
6.2.6，
\(\varphi\)保持线性运算。
