节 6.3 线性映射
子节 6.3.1 基础知识回顾
命题 6.3.2.
设\(V\)和\(U\)为数域\(\F\)上的两个线性空间，\(\varphi : V \to U\)为集合\(V\)到\(U\)的一个映射。\(\varphi\)是线性映射当且仅当，对于任意\(\alpha, \beta \in V\)以及\(c_{1}, c_{2} \in \F\)有
\begin{equation}
\varphi(c_{1} \alpha + c_{2} \beta) = c_{1} \varphi(\alpha) + c_{2} \varphi(\beta).\tag{6.3.1}
\end{equation}
命题 6.3.3.
设\(V\)和\(U\)为数域\(\F\)上的两个线性空间，若映射\(\varphi : V \to U\)是线性映射，则
\begin{equation*}
\varphi(0_{V}) = 0_{U},
\end{equation*}
其中\(0_{V}\)为\(V\)空间中的零向量，\(0_{U}\)为\(U\)空间中的零向量。
例 6.3.4. 恒等映射.
练习 6.3.2 练习
基础题.
1.
判断下面的映射是不是线性映射？
-
\(\varphi: \F^{3} \to \F^{3}, (x,y,z)^{T} \mapsto (x+3y, y-z, z+4x)^{T}\)；
-
\(\varphi: \F^{3} \to \F^{3}, (x,y,z)^{T} \mapsto (xy, yz, zx)^{T}\)；
-
\(\varphi: V \to V, \alpha \mapsto \alpha + \alpha_{0}\)，其中\(V\)是线性空间，\(\alpha_{0} \in V\)是预先给定的一个向量；
-
\(\varphi: \F^{n} \to \F^{m}, \alpha \mapsto A \alpha + \beta\)，其中\(A \in \F^{m \times n}, \beta \in \F^{m}\)；
解答.
-
是线性映射。任取\(\F^{3}\)中向量\((x_{1},y_{1},z_{1})^{T}\)和\((x_{2}, y_{2}, z_{2})^{T}\)，以及\(c_{1}, c_{2} \in \F\)，有\begin{align*} \amp c_{1} (x_{1}, y_{1},z_{1})^{T} + c_{2} (x_{2},y_{2},z_{2})^{T} \\ = \amp (c_{1} x_{1} + c_{2} x_{2}, c_{1} y_{1} + c_{2} y_{2}, c_{1} z_{1} + c_{2} z_{2})^{T}. \end{align*}根据\(\varphi\)的定义有\begin{align*} \amp \varphi(c_{1} (x_{1}, y_{1},z_{1})^{T} + c_{2} (x_{2},y_{2},z_{2})^{T})\\ = \amp \begin{pmatrix} c_{1} x_{1} + c_{2} x_{2}+ 3 c_{1} y_{1} + 3 c_{2} y_{2}\\ c_{1} y_{1} + c_{2} y_{2} - c_{1} z_{1} - c_{2} z_{2}\\ c_{1} z_{1} + c_{2} z_{2} + 4 c_{1} x_{1} + 4 c_{2} x_{2} \end{pmatrix}. \end{align*}将等式左侧重新组织一下有\begin{align*} \amp \varphi(c_{1} (x_{1}, y_{1},z_{1})^{T} + c_{2} (x_{2},y_{2},z_{2})^{T})\\ = \amp c_{1}\begin{pmatrix} x_{1} + 3y_{1}\\ y_{1} - z_{1}\\ z_{1} + 4 x_{1} \end{pmatrix} + c_{2}\begin{pmatrix} x_{2} + 3y_{2}\\ y_{2} - z_{2}\\ z_{2} + 4 x_{2} \end{pmatrix}\\ = \amp c_{1} \varphi((x_{1}, y_{1}, z_{1})^{T}) + c_{2} \varphi((x_{2},y_{2},z_{2})^{T}) \end{align*}根据命题6.3.2，\(\varphi\)是线性映射。
-
不是线性映射。例如，取\(c=2\)，向量\(\alpha = (1,1,1)^{T}\)，则\(\varphi(c \alpha) = \varphi((2,2,2)^{T}) = (4,4,4)^{T}\)，而\(c\varphi(\alpha) = 2(1,1,1)^{T} = (2,2,2)^{T}\)，两者不相等。\(\varphi\)不保持数乘运算，因而不是线性映射。
-
不一定是线性映射。只有当\(\alpha_{0} = 0\)时才是线性映射。因为若\(\alpha_{0} \neq 0\)，则\(\varphi(0) = \alpha_{0} \neq 0\)，而线性映射必须将零向量映到零向量（命题6.3.3）。此外，我们也可以验证\(\varphi\)不保持加法运算：\(\varphi(\alpha+\beta) = \alpha_{0}\)，但\(\varphi(\alpha)+\varphi(\beta)=\alpha_{0}+\alpha_{0}=2\alpha_{0}\)，除非\(\alpha_{0}=0\)，否则不相等。
-
不一定是线性映射（除非\(\alpha_{0}=0\)）。因为，当时\(\alpha_{0} \neq 0\)，\(\varphi(0) = 0 + \alpha_{0} = \alpha_{0} \neq 0\)，不满足线性映射保零的性质（命题6.3.3）。此外，也可以验证\(\varphi\)不保持加法运算：\(\varphi(\alpha+\beta) = \alpha+\beta+\alpha_{0}\)，而\(\varphi(\alpha)+\varphi(\beta) = (\alpha+\alpha_{0})+(\beta+\alpha_{0})=\alpha+\beta+2\alpha_{0}\)。当\(\alpha_{0} \neq 0\)时，两者不等。
-
不一定是线性映射（除非\(\beta=0\)）。类似于上一题，因为\(\varphi(0)=A0+\beta=\beta\)，若\(\beta\neq0\)，则\(\varphi(0)\neq0\)，不满足线性映射保零的性质（命题6.3.3）。只有当\(\beta=0\)时，它是线性映射（即矩阵乘法变换）。
-
不是线性映射。因为取\(c={\rm i } \in \C\)，\(\alpha=1\)，则\(\varphi(c \alpha)=\varphi({\rm i })=\overline{{\rm i }}=-{\rm i }\)，而\(c\varphi(\alpha)={\rm i }\)，两者不相等，\(\varphi\)不能保持数乘运算。注意：若将\(\C\)视为\(\R\)上的线性空间，则该映射是线性的，请自行验证。
2.
设\(V\)为数域\(\F\)上的线性空间，给定\(c \in \F\)，考虑\(V\)到\(V\)的映射\(\varphi: \alpha \mapsto c \alpha\)，证明：\(\varphi\)为线性变换。（形如\(\varphi\)的线性变换被称为数乘变换）
解答.
对任意\(\alpha, \beta \in V\)和任意\(k \in \F\)，有
\begin{align*}
\varphi(\alpha + \beta)\amp = c(\alpha + \beta)\\
\amp = c\alpha + c\beta\\
\amp = \varphi(\alpha) + \varphi(\beta),\\
\varphi(k\alpha) \amp
= c(k\alpha) = (ck)\alpha = (kc)\alpha = k(c\alpha) = k\varphi(\alpha).
\end{align*}
根据定义，\(\varphi\)是线性变换。
3.
给定矩阵\(A \in \F^{m \times n}, B \in \F^{n \times m}\)，证明：
\begin{equation*}
\varphi: \F^{n \times n}\to \F^{m \times m}, X \mapsto AXB
\end{equation*}
是线性映射。
4.
考虑连续函数空间\(C[a,b]\)到自身的映射\(\varphi: f \mapsto \int_{a}^{x} f(t) dt\)，判断\(\varphi\)是否是\(C[a,b]\)上的线性变换。
解答.
是线性变换。因为对任意\(f, g \in C[a,b]\)和任意\(c \in \R\)，有
\begin{align*}
\varphi(f+g)(x) \amp = \int_a^x (f(t)+g(t)) dt\\
\amp = \int_a^x f(t) dt + \int_a^x g(t) dt\\
\amp = \varphi(f)(x) + \varphi(g)(x), \quad \forall x \in [a,b]\\
\varphi(c f)(x) \amp =\int_a^x c f(t) dt \\
\amp= c \int_a^x f(t) dt \\
\amp = c \varphi(f)(x), \quad \forall x \in [a,b].
\end{align*}
所以\(\varphi(f+g) = \varphi(f) + \varphi(g)\)，\(\varphi(cf) = c\varphi(f)\)，满足线性变换的定义。
提高题.
5.
设\(V\)是数域\(\F\)的线性空间，\(\dim V=1\)，\(\phi\in\mathcal{L} (V,V)\)。证明：存在\(c\in\F\)，使得对任意\(\alpha\in V\)，都有\(\phi(\alpha)=c\alpha\)。
解答.
取\(\alpha_{0}\ne 0\)，\(\alpha_{0}\)是\(V\)的基（\(\because\dim V=1\)）。则存在唯一\(c\in \F\)使得\(\phi(\alpha_{0})=c \alpha_{0}\)。
对\(\forall \alpha\in V\)，存在\(c_{0}\)，\(\alpha=c_{0} \alpha_{0}\)，
\begin{equation*}
\phi(\alpha)=\phi(c_{0} \alpha_{0})=c_{0}\phi(\alpha_{0})=c_{0}c \alpha_{0}=c \alpha.
\end{equation*}
6.
设\(V,U\)是有限维线性空间， \(V'\)是\(V\)的子空间。证明：对于任意\(V'\)到\(U\)的线性映射\(\varphi' \in \mathcal{L}(V',U)\)，总存在\(V\)到\(U\)的线性映射\(\varphi \in \mathcal{L}(V,U)\)使得\(\varphi(\alpha) = \varphi'(\alpha), \forall \alpha \in V'\)。（换言之，子空间\(V'\)上的线性映射\(\varphi'\)总可以推广成全空间\(V\)的线性映射）
解答.
取\(V'\)的一组基\((\xi_{1},\ldots,\xi_{k})\)，并将其扩充为\(V\)的一组基\((\xi_{1},\ldots,\xi_{k},\xi_{k+1},\ldots,\xi_{n})\)（参考定理4.4.1）。对任意\(\alpha \in V\)，存在唯一坐标\((a_{1},\ldots,a_{n})^{T}\)使得\(\alpha = \sum_{i=1}^{n} a_{i} \xi_{i}\)。
\begin{equation*}
\varphi(\alpha) = \sum_{i=1}^{k} a_{i} \varphi'(\xi_{i}) + \sum_{i=k+1}^{n} a_{i} \cdot 0,
\end{equation*}
下面首先说明\(\varphi\)是线性映射。
对于任意\(\beta = \sum_{i=1}^{n} b_{i} \xi_{i} \in V\)（其中\((b_{1}, \ldots, b_{n})^{T}\)是\(\beta\)在\((\xi_{1}, \ldots, \xi_{n})\)下的坐标），以及任意\(c_{1}, c_{2} \in \F\)有
\begin{equation*}
\varphi(c_{1} \alpha + c_{2} \beta) = \varphi( \sum_{i=1}^{n} (c_{1} a_{i} + c_{2} b_{i}) \xi_{i} ) = \sum_{i=1}^{k} (c_{1} a_{i} + c_{2} b_{i}) \varphi'(\xi_{i}).
\end{equation*}
对等式右边重新进行组织之后有
\begin{equation*}
\varphi(c_{1} \alpha + c_{2} \beta) = c_{1} \sum_{i=1}^{k} a_{i} \varphi'(\xi_{i}) + c_{2} \sum_{i=1}^{k} b_{i} \varphi'(\xi_{i}) = c_{1} \varphi(\alpha) + c_{2} \varphi(\beta).
\end{equation*}
最后说明，\(\varphi\)与\(\varphi'\)在\(V'\)上一致。当\(\alpha \in V'\)时，\(\alpha\)可表示为\(\sum_{i=1}^{k} x_{i} \xi_{i}\)，此时\(\varphi(\alpha) = \sum_{i=1}^{k} x_{i} \varphi'(\xi_{i}) = \varphi'(\alpha)\)。因此\(\varphi\)是\(\varphi'\)的扩张。
7.
设\(\varphi: V \to U\)是从线性空间\(V\)到线性空间\(U\)的线性映射，证明：若\(U'\)是\(U\)的子空间，则\(\{ \alpha \in V \mid \varphi(\alpha) \in U' \}\)是\(V\)的子空间。
解答.
记\(W = \{ \alpha \in V \mid \varphi(\alpha) \in U' \}\)。
-
对任意\(\alpha, \beta \in W\)，有\(\varphi(\alpha) \in U'\)，\(\varphi(\beta) \in U'\)。由于\(U'\)是子空间，\(\varphi(\alpha)+\varphi(\beta) \in U'\)。由于\(\varphi\)保持加法运算，有\(\varphi(\alpha+\beta)=\varphi(\alpha)+\varphi(\beta)\)，所以\(\alpha+\beta \in W\)。
-
对任意\(\alpha \in W\)有\(\varphi(\alpha) \in U'\)。因为\(U'\)是子空间，所以对任意\(c \in \F\)有\(c\varphi(\alpha) \in U'\)。由于\(\varphi\)保持数乘运算，\(\varphi(c\alpha)=c\varphi(\alpha)\)，所以\(c\alpha \in W\)。
8.
设\(V\)为线性空间，向量组\(\alpha_{1}, \ldots, \alpha_{k} \in V\)线性相关，又设\(U\)是维数大于等于\(1\)的线性空间。证明：可以找到\(U\)中的\(k\)个向量\(\beta_{1}, \ldots, \beta_{k} \in U\)使得，不存在\(V\)到\(U\)的线性映射\(\varphi \in \mathcal{L}(V,U)\)满足\(\varphi(\alpha_{i}) = \beta_{i}, \forall i \in [k]\)。
解答.
由于\(\alpha_{1}, \ldots, \alpha_{k}\)线性相关，存在不全为零的标量\(c_{1}, \ldots, c_{k}\)使得
\begin{equation*}
c_{1} \alpha_{1} + \cdots + c_{k} \alpha_{k} = 0.
\end{equation*}
不妨设\(c_{j} \neq 0, j \in [k]\)。因为\(\dim U \geq 1\)，可取非零向量\(\beta \in U\)。定义\(U\)中的向量
\begin{equation*}
\beta_{i} = \begin{cases}\beta, & i = j, \\ 0, & i \neq j.\end{cases}
\end{equation*}
下面用反证法，假设存在线性映射\(\varphi: V \to U\)满足\(\varphi(\alpha_{i}) = \beta_{i}\)对所有\(i\)成立。则
\begin{align*}
\varphi(c_{1} \alpha_{1} + \cdots + c_{k} \alpha_{k}) \amp = c_{1} \varphi(\alpha_{1}) + \cdots + c_{k} \varphi(\alpha_{k})\\
\amp = c_{1} \beta_{1} + \cdots + c_{k} \beta_{k}\\
\amp = c_{j} \beta.
\end{align*}
但\(c_{1} \alpha_{1} + \cdots + c_{k} \alpha_{k} = 0\)，根据命题6.3.3有\(\varphi(0)=0\)，于是\(c_{j} \beta = 0\)。由于\(c_{j} \neq 0\)，所以必然有\(\beta=0\)。这与已知条件\(\beta \neq 0\)矛盾。因此这样的线性映射\(\varphi\)不存在。
