线性函数

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 2.2 向量空间

给定两个向量空间 $V$ 与 $W$，若映射 $f:V\to W$ 对任意 $a_1,a_2\in K$ 和 $v_1,v_2\in V$ 都满足 $f(a_1{v}_1+a_2{v}_2)=a_{1}f(v_1)+a_{2}f(v_2)$，则称它为线性映射(linear map)。线性映射是保持向量加法与标量乘法的同态的一个例子。$f$ 的像(image)为 $f(V)\subset W$，$f$ 的核(kernel)为 $\{v\in V|f(v)=0\}$，二者分别记作 $\mathrm{im}f$ 和 $\ker f$。$\ker f$ 不可能是空集，因为 $f(0)$ 总是 $0$。若 $W$ 就是域 $K$ 本身，则称 $f$ 为线性函数(linear function)。若 $f$ 是同构，则称 $V$ 与 $W$ 是同构的(isomorphic)，反之亦然，记作 $V\cong W$。由此可得 $\dim V=\dim W$。事实上，所有 $n$ 维向量空间都与 $K^n$ 同构，因而被视为同一个向量空间。向量空间之间的同构是 $\mathrm{GL}(n,K)$ 的元素。

英文原文

Given two vector spaces $V$ and $W$ , a map $f:V\to W$ is called a linear map if it satisfies $f(a_1{v}_1+a_2{v}_2)=a_{1}f(v_1)+a_{2}f(v_2)$ for any $a_1,a_2\in K$ and $v_1,v_2\in V$ . A linear map is an example of a homomorphism that preserves the vector addition and the scalar multiplication. The image of $f$ is $f(V)\subset W$ and the kernel of $f$ is $\{v\in V|f(v)=0\}$ and denoted by $\mathrm{im}f$ and $\ker f$ respectively. $\ker f$ cannot be empty since $f(0)$ is always $0$ . If $W$ is the field $K$ itself, $f$ is called a linear function. If $f$ is an isomorphism, $V$ is said to be isomorphic to $W$ and vice versa, denoted by $V\cong W$ . It then follows that $\dim V=\dim W$ . In fact, all the $n$ -dimensional vector spaces are isomorphic to $K^n$ , and they are regarded as identical vector spaces. The isomorphism between the vector spaces is an element of $\mathrm{GL}(n,K)$ .

出处：2.2 向量空间 · node a-090fa464b4c47451eaabadf9 · 源节点 SHA-256 819d3a11524ad1f7…