102 lines
3.2 KiB
Markdown
102 lines
3.2 KiB
Markdown
# Lecture 13
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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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### Matrices 3C
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#### Theorem 3.63
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A linear map is invertible if and only if it is injective and surjective.
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#### Example
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Consider $T:\mathscr{P}(\mathbb{F})\to \mathscr{P}(\mathbb{F})$, $T(f)=xf$
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$T$ is injective but not surjective. Since you cannot get constant from multiply $x$. So it is not invertible.
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#### Theorem 3.65
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Let $V$ and $W$ be finite-dimensional with the same dimension, and $T\in\mathscr{L}(V,W)$, then $T$ is invertible, if and only if $T$ is injective if and only if, $T$ is surjective.
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Proof:
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Suppose $T$ is injective, then $null\ T={0}$, i.e $dim(null\ T)=0$, since $dim\ V=dim\ null\ T+dim\ range\ T$, we have $dim\ V=dim\ range\ T$ but $dim\ V\dim\ W$, so $dim\ W=dim\ range\ T$. Thus $W=range\ T$. This shows that $T\ injective \implies T\ surjective$.
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If $T$ is surjective, then $dim\ range\ T=dim\ W$ but then $dim\ V=dim\ null\ T+dim\ W\implies dim\ null\ T=0$, so $T$ is injective, $T\ surjective\implies T\ injective$.
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#### Theorem 3.68
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Suppose $V,W$ finite dimensional $dim\ V=dim\ W$, then for $T\in\mathscr{L}(V,W)$ and $S\in \mathscr{L}(W,V)$, then $ST=I\implies TS=I$
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#### Example 3.67
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Show that for a polynomial $q$ with degree $m$, there exists a unique polynomial $p$ of degree $m$ such that $((x^2+5x+7)p)''=q$
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Solution:
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Let $T:\mathscr{P}_m(\mathbb{F})\to \mathscr{P}_m(\mathbb{F})$ given by $T(p)=((x^2+5x+7)p)''$ by $T$ is injective since $(x^2+5x+7)$ has degree $\geq 2$ for $p\neq 0$, therefore, $p$ is surjective. (by **Theorem 3.68**)
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#### Isomorphisms
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#### Definition 3.69
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An **isomorphism** of vector spaces is a invertible linear map. Two vector spaces $V,W$ are isomorphic if there exists an isomorphism between them.
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Notation: $V\cong W$ means $V$ and $W$ are isomorphic. (Don't use very often, no map is included.)
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Example:
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$\mathscr{P}_m(\mathbb{F})$ and $\mathbb{F}^{m+1}$ are isomorphic. $T:\mathbb{F}^{m+1}\to \mathscr{P}_m(\mathbb{F}): T((a_0,...,a_m))=a_0+a_1x+...+a_n x^n$
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#### Theorem 3.70
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Two finite dimensional vector spaces $V,W$ are isomorphic if and only if $dim\ V= dim\ W$
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Ideas of Proof:
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$\Rightarrow$ use fundamental theorems of linear map
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$\Leftarrow$ Let $v_1,...,v_m\in V$ and $w_1,...,w_n\in W$ be bases. Then define $T:V\to W$ by $T(v_k)=w_k$ for $1\leq k\leq n$
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Show $T$ is invertible by showing $T$ is injective and surjective.
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#### Theorem 3.71
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Let $V,W$ be finite dimensional, let $v_1,...,v_n\in V$ and $w_1,...,w_m\in W$ be bases. Then the map
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$$
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M(-,(v_1,...,v_n),(w_1,...,w_m)):\mathscr{L}(V,W)\to \mathbb{F}^{m,n}
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$$
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$T\mapsto M(T)$ or $M(-,(v_1,...,v_n),(w_1,...,w_m))$ is an isomorphism ($M:\mathscr{L}(V,W)\to \mathbb{F}^{m,n}$)
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Sketch of Proof:
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Need to show $M$ is surjective and injective.
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* Injective: i.e need to show if $M(T)=0$, then $T=0$. $M(T)=0\implies Tv_k=0, 1\leq k\leq n$
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* Surjective: i.e let $A\in F^{m,n}$ define $T:V\to W$ given by $Tv_k=\sum_{j=1}^m A_{j,k} w_j$ you cna check that $M(T)=A$
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#### Corollary 3.72
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$dim \mathscr{L}(V,W)=(dim\ V)(dim\ W)$
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#### Definition 3.73
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$v\in V, v_1,...,v_n$ a basis, then $M(v)=\begin{pmatrix}
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b_1\\
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...\\
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b_n
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\end{pmatrix}, v=a_1v_1,...,a_nv_n$
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#### Proposition 3.75, 3.76
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$$
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M(T)_{\cdot,k}=M(Tv_k)
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$$
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$$
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M(Tv)=M(T)M(v)
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$$ |