4.1 线性映射的概念

映射

\(f:A \to B\)
像:\(f:a \mapsto b, b=f(a),a\)为原像
像集:\(Imf=f(A):=\{f(a)|a\in A\}\)
满射:\(f(A)=B\),像集是B全体
单射:\(a_1\neq a_2\in A\Rightarrow f(a_1)\neq f(a_2)\),原像不同,像不同
\(f(a_1)= f(a_2)\in f(A)\Rightarrow a_1= a_2\in A\),像同,原像同

双射:即单又满

复合:\(f:A\to B,g:B\to C,h:C\to D\),则
\(g\circ f:=g(f(a))\)

复合:\(f:A\to B,g:B\to C,h:C\to D\),则
\((h\circ g)\circ f=h\circ (g\circ f)\)
\(\forall a\in A, ((h\circ g)\circ f)(a) = (h\circ g)(f(a))=h(g(f(a)))\)
\(\forall a\in A, (h\circ (g\circ f))(a) = h((g\circ f)(a))=h(g(f(a)))\)

逆映射:\(f:A \to B\)为双射,则\(g:B \to A,gf=1_A,fg=1_B,g=f^{-1}\)

命题4.1.1:设\(f\)是集合\(A\to B\)的映射,如果\(\exists B\to A\)的映射\(g\)\(s.t.gf=1_A,fg=1_B,\)
\(f\)是双射,且\(g=f^{-1}\)

满:\(\forall b\in B,由g:B\to A,\exists a=g(b)\in A, s.t. f(a)=f(g(b))=(fg)(b)=1_B(b)=b\)
单:取\(由f:A\to B,取a_1,a_2\in A, f(a_1)=f(a_2)\in f(A),\)
\(a_1=1_A(a_1)=(gf)(a_1)=(g(f(a_1))=(g(f(a_2))=(gf)(a_2)=1_A(a_2)=a_2\)

线性映射

定义4.1.1\(\varphi\)是数域\(K\)上线性空间\(V\to U\)的映射,如果\(\varphi\)适合下列条件:

  1. \(\varphi(\alpha+\beta)=\varphi(\alpha)+\varphi(\beta),\alpha,\beta\in V\)
  2. \(\varphi(k\alpha)=k\varphi(\alpha),k\in K,\alpha\in V\)

则称\(\varphi是V\to U\)的线性映射。\(V\)到自身的线性映射称为\(V\)上的线性变换。若\(\varphi:V\to U\)是单的,则称\(\varphi\)是单线性映射;若\(\varphi\)是满的,则称\(\varphi\)是满线性映射,若\(\varphi\)是双射,则称\(\varphi\)是线性同构(同构映射),简称同构。若\(V=U,V\)自身上的同构称为自同构。

命题4.1.2:设\(\varphi是V\to U\)的线性映射,则:

  1. \(\varphi(0_V)=0_U\);
  2. \(\varphi(k\alpha+l\beta)=k\varphi(\alpha)+l\varphi(\beta),\alpha,\beta\in V,k,l\in K\);
  3. \(\varphi\)同构,则其逆映射\(\varphi^{-1}\)也是线性映射,从而是\(U\to V\)的同构。
posted @ 2020-12-24 18:35  一花一世界,一叶一乾坤  阅读(1751)  评论(0编辑  收藏  举报