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content/Math429/Math429_L8.md

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+# Lecture 8
+
+## Chapter III Linear maps
+
+**Assumption: $U,V,W$ are vector spaces (over $\mathbb{F}$)**
+
+### Vector Space of Linear Maps 3A
+
+#### Definition 3.1
+
+A **linear map** from $V$ to $W$ is a function from $T:V\to W$ with the following properties:
+
+1. Additivity: $T(u+v)=T(u)+T(v),\forall u,v\in V$
+2. Homogeneity: $T(\lambda v)=\lambda T(v),\forall \lambda \in \mathbb{F},v\in V$
+
+#### Notation
+
+* $Tv=T(v)$
+* $\mathscr{L}(V,W)$ denotes the set of linear maps from $V$ to $W$. (homomorphism, $Hom(V,W)$)
+* $\mathscr{L}(V)$ denotes the set of linear maps from $V$ to $V$. (endomorphism, $End(V)$)
+
+#### Example
+
+* zero map $0(v)\in \mathscr{L}(V,W)$ $0(v)=0$
+* identity map $I\in \mathscr{L}(V,W)$, $I(v)=v$
+* scaling map $T\in \mathscr{L}(V,W)$, $T(v)=av,a\in \mathbb{F}$
+* differentiation map $D\in \mathscr{L}(\mathscr{P}_m(\mathbb{F}),\mathscr{P}_{m-1}(\mathbb{F}))$, $D(f)=f'$
+
+#### Lemma 3.10
+
+$T0=0$ for $T\in \mathscr{L}(V,W)$
+
+Proof:
+
+$T(0+0)=T(0)+T(0)$
+
+
+#### Theorem 3.4 Linear map lemma
+
+Suppose $v_1,...,v_n$ is a basis for $V$, and suppose $w_1,...,w_n\in W$ are arbitrary vector. Then, there exists a unique linear map. $T:V\to W$ such that $T_{v_i}=w_i$ for $i=1,...,n$
+
+Proof:
+
+First we show existence.
+
+by constrains,
+
+$T(c_1 v_1,...+c_n v_n)=c_1w_1+...+c_n  w_n$
+
+T is well defined because $v_1,....v_n$ are a basis.
+
+Need to show that $T$ is a linear map.
+
+* Additivity: let $u,v\in V$ and suppose $a_1,...,a_n,b_1,...,b_n\in \mathbb{F}$
+with $u=a_1v_1+....+a_n v_n ,v=b_1v_1+...+b_2v_n$, then $T(u+v)=T((a_1+b_1)v_1+...+(a_n+b_n)v_n)=Tu+Tv$
+
+Proof for homogeneity used for exercise.
+
+Need to show $T$ is unique. Let $S\in\mathscr{L}(V,W)$ such that $Sv_i=w_i,i=1,...,n$
+
+$$
+S(c_1 v_1+...+c_n v_n)=S(c_1v_1)+S(...)+S(c_n v_n)=c_1S(v_1)+...+c_nS(v_n)
++c_1w_1+...+c_nw_n
+$$
+
+Then $S=T$
+
+#### Definition 3.5
+
+Let $S,T\in \mathscr{L}(V,W)$, then define
+
+* $(S+T)\in\mathscr{L}(V,W)$ by $(S+T)(v)=Sv+Tv$
+* for $\lambda \in \mathbb{F}$, $(\lambda T)\in \mathscr{L}(V,W)$, $(\lambda T)(v)=\lambda T(v)$
+
+Exercises: Show that $S+T$ and $\lambda T$ are linear maps.
+
+#### Theorem 3.6
+
+$\mathscr{L}(V,W)$ is a vector space.
+
+Sketch of proof:
+
+* additive identity: $0(v)=0$
+* associativity:
+* commutativity:
+* additive inverse: $T\to (-1)T=-T$
+* scalar multiplication $1T=T$
+* distributive
+
+#### Definition 3.7
+
+Multiplication for linear map: $(ST)v=S(T(v))=(S\circ T)(v)$ **Not commutative but associative**.