Fundamental Theorem of Linear Algebra

Definition

Let $\mathcal{L}(V,W):=\{L:V\to W\}$ where $L$ is linear map and define a sum and scalar operation to the $\mathcal{L}(V,W)$

• $(L_1+L_2)(\mathbf{v})=L_1(\mathbf{v})+L_2(\mathbf{v})$
• $(kL)(\mathbf{v})=kL(\mathbf{v})$

Where $L\in \mathcal{L}(V,W)$, $\mathbf{v}\in V$, $k\in F$, and ${\mathcal{M}}_{m\times n}$is a set of $m\times n$ matrices on $F$

Let the functions $f,g$

• $f:\mathcal{L}(V,W)\to {\mathcal{M}}_{m\times n}(F)$ $f(L)=[L{]}_{B_W}^{B_V}=M$
• $g:{\mathcal{M}}_{m\times n}(F)\to \mathcal{L}(V,W)$ $g(M)=L_M$ where $L_M$ satisfies $[L_M(\mathbf{v}){]}_{B_W}=M[\mathbf{v}{]}_{B_V}$

And terms

• $[\mathbf{v}{]}_{B_V}:=(k_1,\dots ,k_n{)}^{⊺}$ for $\mathbf{v}=(k_1{\mathbf{v}}_1+\cdots +k_n{\mathbf{v}}_n)\in V$
• $[L{]}_{B_W}^{B_V}:=([L({\mathbf{v}}_1){]}_{B_W},\dots ,[L({\mathbf{v}}_n){]}_{B_W})$

where $L(\mathbf{v})=(l_1{\mathbf{w}}_1+\cdots +l_n{\mathbf{w}}_n)\in W$, and $[L(\mathbf{v}){]}_{B_W}=(l_1,\dots ,l_n{)}^{⊺}$ is a column vector. and $B_V,B_W$ are the ordered basis of $V,W$ respectively

Then, $f,g$ both are Isomomorphism and the inverse maps of each other.

If $T:V\to W$, then $\exists A$ s.t. $\forall \mathbf{v}\in V:T(\mathbf{v})=A\mathbf{v}$ Every Linear Map between finite-demensional vector spaces can be represented by a Matrix.

Proof

$f$ is a Linear Map

Proof of Additivity of $f$

$f(L_1+L_2)=f(L_1)+f(L_2)$

$\forall i\in {\mathbb{N}}^{+},\forall L_1,L_2\in \mathcal{L}(V,W):(L_1+L_2)({\mathbf{v}}_i)=L_1({\mathbf{v}}_i)+L_2({\mathbf{v}}_i)$ by the definition of $\mathcal{L}(V,W)$. So, $[(L_1+L_2)({\mathbf{v}}_i){]}_{B_{\mathbf{w}}}=[L_1({\mathbf{v}}_i)+L_2({\mathbf{v}}_i){]}_{B_{\mathbf{w}}}\Rightarrow [(L_1+L_2)({\mathbf{v}}_i){]}_{B_{\mathbf{w}}}=[L_1({\mathbf{v}}_i){]}_{B_W}+[L_2({\mathbf{v}}_i){]}_{B_W}$ by the definition of $f(L)$ $\iff [(L_1+L_2)(\mathbf{v}){]}_{B_W}=[L_1(\mathbf{v}){]}_{B_W}+[L_2(\mathbf{v}){]}_{B_W}$ $\therefore f(L_1+L_2)=f(L_1)+f(L_2)$

Proof of Homogeneity of $f$

$f(kL)=kf(L)$

$\forall k\in F,\forall i\in {\mathbb{N}}^{+},L\in \mathcal{L}(V,W):(kL)({\mathbf{v}}_i)=kL({\mathbf{v}}_i)$ by the definition of $\mathcal{L}(V,W)$. So, $[(kL)({\mathbf{v}}_i){]}_{B_{\mathbf{w}}}=[kL({\mathbf{v}}_i){]}_{B_{\mathbf{w}}}\Rightarrow [(kL)({\mathbf{v}}_i){]}_{B_{\mathbf{w}}}=k[L({\mathbf{v}}_i){]}_{B_{\mathbf{w}}}$ by the definition of $f(L)$ $\iff [(kL)(\mathbf{v}){]}_{B_{\mathbf{w}}}=k[L(\mathbf{v}){]}_{B_{\mathbf{w}}}$ $\therefore f(kL)=kf(L)$

$f$ is a Isomomorphism

Proof of Monomorphism of $f$

$f(L_1)=f(L_2)\Rightarrow L_1=L_2$

$\forall i\in {\mathbb{N}}^{+},\forall L_1,L_2\in \mathcal{L}(V,W):f(L_1)=f(L_2)\iff [L_1({\mathbf{v}}_i){]}_{B_W}=[L_2({\mathbf{v}}_i){]}_{B_W}\iff L_1({\mathbf{v}}_i)=L_1({\mathbf{v}}_i)$ Let $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}=B_V$, then $\forall \mathbf{v}\in V:\exists k_1,\dots ,k_n\in F$ s.t. $\mathbf{v}={\sum }_{i=1}^n{k}_i{\mathbf{v}}_i$ Since $L_1({\mathbf{v}}_i)=L_1({\mathbf{v}}_i)$, ${\sum }_{i=1}^n{k}_i{L}_1({\mathbf{v}}_i)={\sum }_{i=1}^n{k}_i{L}_2({\mathbf{v}}_i)\iff {\sum }_{i=1}^n{L}_1(k_i{\mathbf{v}}_i)={\sum }_{i=1}^n{L}_2(k_i{\mathbf{v}}_i)\iff L_1({\sum }_{i=1}^n{k}_i{\mathbf{v}}_i)=L_2({\sum }_{i=1}^n{k}_i{\mathbf{v}}_i)$ by the definition of $\mathcal{L}(V,W)$. $\therefore L_1(\mathbf{v})=L_2(\mathbf{v})\iff L_1=L=2$

Proof of Epimorphism of $f$

$f(L)=M$

Let $\forall M\in {\mathcal{M}}_{m\times n}(F):[M{]}^i:=i\text{-th column of }M$, and $\forall i\in {\mathbb{N}}^{+}:[L({\mathbf{v}}_i){]}_{B_W}:=[M{]}^i$ Then, $[L({\mathbf{v}}_i){]}_{B_W}^{B_V}=M$ $\therefore f(L)=M$

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$g$ is a Linear Map

Proof of Additivity of $g$

$g(M_1+M_2)=g(M_1)+g(M_2)$

$\forall M_1,M_2\in {\mathcal{M}}_{m\times n}(F),\forall \mathbf{v}\in V:[L_{M_1+M_2}(\mathbf{v}){]}_{B_W}=(M_1+M_2)[\mathbf{v}{]}_{B_V}$ by the definition of the operation $g(M)$ $(M_1+M_2)[\mathbf{v}{]}_{B_V}=M_1[\mathbf{v}{]}_{B_V}+M_2[\mathbf{v}{]}_{B_V}=[L_{M_1}(\mathbf{v}){]}_{B_W}+[L_{M_2}(\mathbf{v}){]}_{B_W}$ by the definition of matrix operation $\therefore g(M_1+M_2)=g(M_1)+g(M_2)$

Proof of Homogeneity of $g$

$g(kM)=kg(M)$

$\forall k\in F,\forall M\in {\mathcal{M}}_{m\times n}(F):[L_{kM}(\mathbf{v}){]}_{B_W}=(kM)[\mathbf{v}{]}_{B_V}=kM[\mathbf{v}{]}_{B_V}=k[L_M(\mathbf{v}){]}_{B_W}$ $\therefore g(kM)=kg(M)$

$g$ is a Isomomorphism

Proof of Monomorphism of $g$

$g(M_1)=g(M_2)\Rightarrow M_1=M_2$

$\forall M_1,M_2\in {\mathcal{M}}_{m\times n}(F),\forall \mathbf{v}\in V:g(M_1)=g(M_2)\iff [L_{M_1}(\mathbf{v}){]}_{B_W}=[L_{M_2}(\mathbf{v}){]}_{B_W}\iff M_1[\mathbf{v}{]}_{B_V}=M_2[\mathbf{v}{]}_{B_V}\iff [M_1{]}^i=[M_2{]}^i\iff M_1=M_2$ $\therefore g(M_1)=g(M_2)\Rightarrow M_1=M_2$

Proof of Epimorphism of $g$

$\therefore g(M)=L$

Let $\forall L\in \mathcal{L}(V,W):M:=([L({\mathbf{v}}_1){]}_{B_W},\dots ,[L({\mathbf{v}}_n){]}_{B_W})$ Then, $\forall i\in {\mathbb{N}}^{+}:[L({\mathbf{v}}_i){]}_{B_W}:=[M{]}^i$ by the definition $\therefore g(M)=L_M=L$

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$f,g$ are the inverse maps of each other.

$\forall L\in \mathcal{L}(V,W),\forall v\in V:(g\circ f)(L)=g(f(L))=g(M)=L_M=L\Rightarrow gf=I$

$\forall M\in {\mathcal{M}}_{m\times n}(F):(f\circ g)(M)=f(g(M))=f(L_M)=f(L)=M\Rightarrow fg=I$