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