4.1 線性映射的概念

映射

\(f:A \to B\)
像：\(f:a \mapsto b, b=f(a)，a\)為原像
像集：\(Imf=f(A):=\{f(a)|a\in A\}\)
滿射：\(f(A)=B\)，像集是B全體
單射：\(a_1\neq a_2\in A\Rightarrow f(a_1)\neq f(a_2)\)，原像不同，像不同
或\(f(a_1)= f(a_2)\in f(A)\Rightarrow a_1= a_2\in A\)，像同，原像同

雙射：即單又滿

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

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

逆映射：\(f:A \to B\)為雙射，則\(g:B \to A,gf=1_A,fg=1_B,g=f^{-1}\)

命題4.1.1：設\(f\)是集合\(A\to B\)的映射，如果\(\exists B\to A\)的映射\(g\)，\(s.t.gf=1_A,fg=1_B,\)
則\(f\)是雙射，且\(g=f^{-1}\)

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

線性映射

定義4.1.1 設\(\varphi\)是數域\(K\)上線性空間\(V\to U\)的映射，如果\(\varphi\)適合下列條件：

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

則稱\(\varphi是V\to U\)的線性映射。\(V\)到自身的線性映射稱為\(V\)上的線性變換。若\(\varphi:V\to U\)是單的，則稱\(\varphi\)是單線性映射；若\(\varphi\)是滿的，則稱\(\varphi\)是滿線性映射，若\(\varphi\)是雙射，則稱\(\varphi\)是線性同構（同構映射），簡稱同構。若\(V=U,V\)自身上的同構稱為自同構。

命題4.1.2：設\(\varphi是V\to U\)的線性映射，則：

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. 若\(\varphi\)同構，則其逆映射\(\varphi^{-1}\)也是線性映射，從而是\(U\to V\)的同構。