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\begin{document}
 \parindent=0cm
\section*{ 5. Tensors as multi-linear functions.}

  $<$Non-singular pairing.$>$

  A pairing of $V$ and $W$ is a bilinear map $(\,\,,\,\,):V\times
  W\rightarrow\rb(or\,\,\cb)$.

  $(\,\,,\,\,)$ is nonsingular if $(1)(v,\forall w)=0\Rightarrow v=0$

  $\hs{9.5em}(2)(\forall v,w)=0\Rightarrow w=0$

  $(1)\Leftrightarrow V\rightarrow W^*$ is 1-1.

  $\hs{3em}v\mapsto(v,\,\,)$

$(2)\Leftrightarrow W\rightarrow V^*$ is 1-1.

  $\hs{3em}w\mapsto(\,\,,w)$

  À§¿¡¼­ $dim\,V\leq dim\,W^*$,$dim\,W\leq dim\,V^*$¸¦ ¾òÀ» ¼ö ÀÖ°í $dim\,V=dim\,V^*$À̹ǷÎ
µû¶ó¼­ $\pr$ÀÌ nonsingularÇÏ´Ù¸é $V,W$´Â °°Àº Â÷¿øÀÓÀ» ¾Ë ¼ö ÀÖ´Ù.
Â÷¿øÀÌ °°°í 1-1À̹ǷΠÀ§ ÇÔ¼öµéÀº isomorphism $V\cong W^*,W\cong
V^*$ À» ÁØ´Ù.

º¸´Ù ÀϹÝÀûÀ¸·Î


$\pr:V\times W\rightarrow \rb$, a pairing with

$(1) (n,\,\,)=0\,\,,\forall n\in N< V$

$(2) (\,\,,m)=0\,\,,\forall m\in M< W$.

Then $\pr$ induces a pairing $\pr:V/N\times
W/M\rightarrow\rb$ given by $([v],[w])$:=(v,w).\\

ÀÌÁ¦ À§ÀÇ ³»¿ëÀ»  tensor¿¡ Àû¿ëÇÏÀÚ.
$M(V^*)=span\{V^*\times\cdot\cdot\times V^*\}$,

$M(V)=span\{V\times\cdot\cdot\times V\}$ ¿¡ ´ëÇØ ´ÙÀ½ pairingÀ»
»ý°¢ÇÑ´Ù.

$(*)\,\,\,M(V^*)\hs{1em}\times
\hs{1em}M(V)\,\,\,\,\,\,\longrightarrow\hs{1em}\rb\hs{6em}$

$\,\,\,\,\,\,\,\,\,\,\,((\alp_1,\cdot\cdot,\alp_r),(v_1,\cdot\cdot,v_r))\,\,\,\,\,\mapsto\,\,\,\alp_1(v_1)\cdot\cdot\alp_r(v_r)$\\

¾Õ¿¡¼­ $V\times\cdot\cdot \times V$¿¡¼­ $T^r(V)$¸¦ ¾òÀ» ¶§ ½è´ø
N¿¡ ´ëÇØ $N(V^*),N(V)$´Â

$((\alp_1+\alp_1',\cdot\cdot)-(\alp_1,\cdot\cdot)-(\alp_1',\cdot\cdot),\forall
(v_1,\cdot\cdot\cdot,v_r) )\mapsto 0$ ,

$(\forall (\alp_1,\cdot\cdot\cdot,\alp_r)
,(v_1+v_1',\cdot\cdot)-(v_1,\cdot\cdot)-(v_1',\cdot\cdot))\mapsto
0$ µîÀ» ¸¸Á·ÇÏ°í µû¶ó¼­

$\pr:T^r(V^*)\times T^r(V)\rightarrow\rb$ ÀÌ
induceµÈ´Ù. Áï\\

$M(V^*)\times M(V)\hs{1em}\rightarrow\hs{1em}\rb$

$\hs{2em}\pi\downarrow\hs{3em}\nearrow \exists\pr$

$T^r(V^*)\times T^r(V)$\\

ÀÌÁ¦ ÀÌ pairingÀÌ nonsingular ÇÏ´Ù´Â °Í¸¸ º¸À̸é $T^r(V^*)\cong
(T^r(V))^*$ÀÓÀ» ¾Ë ¼ö ÀÖ´Ù. $V$ÀÇ basis $e=(e_1,\cdot\cdot\cdot,
e_n)$¿¡ ´ëÇØ $V^*$¿¡¼­ dual basis
$\eps=(\eps_1,\cdot\cdot\cdot,\eps_n)$À» Àâ¾ÒÀ» ¶§ ,
$(\sum_Ia_I\eps_I,\forall v)=0$, $a_I\eps_I=a_{i_1\cdot\cdot
i_r}\eps_{i_1}\ot\cdot\cdot\cdot\ot\eps_{i_r}$À̶ó¸é
$v=e_J=e_{j_1}\ot\cdot\cdot\ot e_{j_r}$¿¡ ´ëÇØ
$0=(\sum_Ia_I\eps_I,e_J)=\sum_Ia_I\eps_{i_1}(e_{j_1})\cdot\cdot\cdot\eps_{i_r}(e_{j_r})=a_J$

ÀÌ µÇ¾î  ¸ðµç $a_J=0$ÀÌ µÇ°í ´Ù¸¥ÂÊ ¿ª½Ã ¸¶Âù°¡ÁöÀ̹ǷΠÀÌ
pairingÀÌ nonsingularÇÔÀ» ¾Ë ¼ö ÀÖ´Ù. Áï $T^r(V^*)\cong
(T^r(V))^*$  ÀÌ´Ù.

 ¶ÇÇÑ $L_r(V)=\{f|f:V\times\cdot\cdot\times
V\rightarrow\rb,multilinear\}$À̶ó µÎ¸é ´ÙÀ½ÀÇ
universal property ·ÎºÎÅÍ $L_r(V)\cong(T^r(V))^*$ÀÓÀ» ¾Ë ¼ö ÀÖ´Ù.\\


$\hs{2em}V\times\cdot\cdot\times V
\hs{1em}\overset{\pi}{\longrightarrow}\hs{1em}V\ot\cdot\cdot\ot
V=T^r(V)$

$\hs{0.5em}multilinear\,\,f\searrow\hs{3em}\swarrow\exists
!\overline{f}\in L(T^r(V),W)\hs{3em}$

$\hs{9em}W$\\

µû¶ó¼­ $L_r(V)\cong(T^r(V))^*\cong T^r(V^*)$.\\

À§ÀÇ ³»¿ëÀ» $\Lam(V)$¿¡µµ Àû¿ëÇÒ ¼ö ÀÖ´Ù. óÀ½¿¡ ÁÖ¾ú´ø pairing
$(*)$¿¡¼­

\begin{math}
((\alp_1,\cdot\cdot,\alp_r),(v_1,\cdot\cdot,v_r))= \left\{
\begin{array}{c}
 det(\alp_i(v_j)) \cdots I\\
 \frac{1}{r!} det(\alp_i(v_j)) \cdots I\!\!I
 \end{array} \right
\end{math}

·Î ÁÖ¸é detÀÇ ¼ºÁú(Çà ȤÀº ¿­ÀÌ °°Àº °ÍÀÌ ÀÖÀ¸¸é det°ªÀº 0ÀÌ
µÈ´Ù.)¿¡
ÀÇÇØ ´ÙÀ½°ú °°Àº quotient°¡ °¡´ÉÇÏ´Ù. Áï\\

$T^r(V^*)\times T^r(V)\hs{1em}\rightarrow\hs{1em}\rb$

$\hs{3.3em}\downarrow\hs{5em}\nearrow \pr=det$

$\Lam^r(V^*)\times \Lam^r(V)$\\

µû¶ó¼­ $\pr:\Lam^r(V^*)\times \Lam^r(V)\rightarrow\rb$ÀÌ ¾ò¾îÁö°í
ÀÌ ¶§ ÀÌ pairingÀÌ nonsingularÇÔÀº À§¿¡¼­ ÇÑ °Í°ú ¸¶Âù°¡Áö·Î º¸ÀÏ
¼ö ÀÖ´Ù.

$A_r(V)=\{f\,\,|\,\,f:V\times\cdot\cdot\times
V\rightarrow\rb,multilinear\,\,,alternating\,\,map\}$ À̶ó µÎ¸é ÀÌ
¿ª½Ã universal property¿¡ ÀÇÇØ $A_r(V)\cong(\Lam^r(V))^*$ÀÓÀ» ¾Ë
¼ö ÀÖ´Ù. i.e.,\\

 $\hs{5em}A_r(V)\cong(\Lam^r(V))^*\cong\Lam^r(V^*)$ .\\

\section*{ 6. View $\Lam^r(V)\subset T^r(V)$.}

´ÙÀ½ exact sequence ¸¦ »ý°¢Çغ¸ÀÚ.

$(*)\hs{2.5em}0\rightarrow \textbf{a}_r\rightarrow
T^r(V)\overset{\pi}{\rightarrow}\Lam^r(V)\rightarrow 0$

Define $s(v_1\we\cdot\cdot\cdot\we
v_r):=\frac{1}{r!}\sum_{\sigma}sgn(\sigma)v_{\sigma(1)}\ot\cdot\cdot\cdot\ot
v_{\sigma(r)} $.

Then $s$ is a section, i.e., $\pi\circ s=id$ so that $T^r(V) = a_r \oplus s(\Lam^r(V))$.\\


$(*)$ ¿¡¼­ $0\leftarrow\textbf{a}_r^*\leftarrow (
T^r(V))^*\overset{\pi^*}{\leftarrow}(\Lam^r(V))^*\leftarrow 0 $
 ¸¦ ¾òÀ» ¼ö ÀÖ°í µû¶ó¼­

 $(\Lam^r(V))^*\underset{\pi^*} {\hookrightarrow}(T^r(V))^* $ ·Î º¼ ¼ö
 ÀÖ´Ù.\\


{\bf ¼÷Á¦ 15.} ´ÙÀ½ diagramÀÌ commuteÇÔÀ» È®ÀÎÇغ¸¶ó.

$L_r(V)\cong (T^r(V))^*\cong T^r(V^*)$

$~~\cup\hs{3.0em}\uparrow\pi^*\hs{3.5em}\uparrow s'$

$A_r(V)\cong (\Lam^r(V))^*\cong \Lam^r(V^*)$\\


\section*{ 7. Algebra structure on $A(V)=\bigoplus_r A_r(V^)$.}

$A_r(V)\cong\Lam^r(V^*)$ ¿¡¼­ algebra structure on $A(V)$´Â $\Lam
(V^*) $ÀÇ algebra structure¿¡¼­ induceµÈ´Ù.

\begin{prop}

For $f\in A_p(V),g\in A_q(V),$

(1) $\dis{f\we_I
g(v_1,\cdot\cdot\cdot,v_{p+q})=\sum_{\sigma=(p,q)shuffle}(sgn
\sigma
)f(v_{\sigma(1)},\cdot\cdot\cdot,v_{\sigma(p)})g(v_{\sigma(p+1)},\cdot\cdot\cdot,v_{\sigma(q)})}$

$\dis{\hs{10.2em}=\frac{1}{p!q!}\sum_{\sigma}(sgn \sigma
)f(v_{\sigma(1)},\cdot\cdot\cdot,v_{\sigma(p)})g(v_{\sigma(p+1)},\cdot\cdot\cdot,v_{\sigma(q)})}$

(2)
$\dis{f\we_{I\!\!I}g(v_1,\cdot\cdot\cdot,v_{p+q})=\frac{1}{(p+q)!}\sum_{\sigma}(sgn
\sigma
)f(v_{\sigma(1)},\cdot\cdot\cdot,v_{\sigma(p)})g(v_{\sigma(p+1)},\cdot\cdot\cdot,v_{\sigma(q)})}$

\end{prop}

(5Àý¿¡¼­ $I , I\!\!I$ ¼³¸í)

\begin{proof}
It suffices to show for
$f=\eps_{i_1}\we\cdot\cdot\cdot\we\eps_{i_p}$,
$g=\eps_{j_1}\we\cdot\cdot\cdot\we\eps_{j_q}$ where
$\eps_1,\cdot\cdot\cdot,\eps_n$ are dual basis of
$e_1,\cdot\cdot\cdot,e_n$ of $V$.

$f\we
g(v_1,\cdot\cdot\cdot,v_{p+q})=(\eps_{i_1}\we\cdot\cdot\cdot\we\eps_{j_q})(v_1,\cdot\cdot\cdot,v_{p+q})=det(\varepsilon(v))$
ÀÌ°í ¾Æ·¡ÀÇ µµ¿òÁ¤¸®·ÎºÎÅÍ Áõ¸íÀÌ ¿Ï¼ºµÈ´Ù. (\bf{¼÷Á¦ 16.})
\end{proof}

\begin{lem}(Lagrange formula)

Given $A=(a_{ij}), n\times n$ matrix and for fixed $p$, let $
A_I(I=i_1<\cdot\cdot\cdot <i_p)$ be a $p\times p$ submatrix given
by $(A_{I})_{kl}=(a_{k,i_l}),k=1,\cdot\cdot\cdot,p$ and
$A_{\hat{I}},\hat{I}=(1,\cdot\cdot\cdot,\hat{i_1},\cdot\cdot,\hat{i_p},\cdot\cdot,n)=(j_1<\cdot\cdot\cdot<
j_q)=J$, be a $q\times q$ submatrix given by
$(A_{\hat{I}})_{kl}=(a_{p+k,j_l})$. Then

$\hs{3em}\dis{det A=\sum_Isgn(I,\hat{I})detA_IdetA_{\hat{I}}}$.

\end{lem}

\begin{proof}
$\dis{detA=\sum_{\sigma}(-1)^{\sigma}a_{1\sigma(1)}\cdot\cdot\cdot
a_{p\sigma(p)}a_{p+1\sigma(p+1)}\cdot\cdot\cdot a_{n\sigma(n)}}$

$\dis{\hs{4.7em}=\sum_{\sigma\in sh(p,q)}\sum_{\rho\in
S_p}\sum_{\tau\in
S_q}(-1)^{\sigma}(-1)^{\rho}(-1)^{\tau}a_{1,\rho\sigma(1)}\cdot\cdot\cdot
a_{p,\rho\sigma(p)}a_{p+1,\tau\sigma(p+1)}\cdot\cdot\cdot
a_{n,\tau\sigma(n)}}$

$\dis{\hs{4.7em}=\sum_{\sigma\in sh(p,q)}\sum_{\tau\in
S_q}(-1)^\sigma(-1)^\tau(\sum_{\rho\in
S_p}(-1)^{\rho}a_{1,\rho\sigma(1)}\cdot\cdot\cdot
a_{p,\rho\sigma(p)})a_{p+1,\tau\sigma(p+1)}\cdot\cdot\cdot
a_{n,\tau\sigma(n)}}$

$\dis{\hs{4.7em}=\sum_{\sigma\in sh(p,q)}(-1)^\sigma detA_I
detA_{\hat{I}}}$
\end{proof}

\begin{prop}
Let $f\in A_p(V)$ and $ g\in A_q(V)$, then $f\we g=(-1)^{pq}g\we
f$.
\end{prop}

\begin{proof}
$f=\eps_{i_1}\we\cdot\cdot\cdot\we\eps_{i_p}$,
$g=\eps_{j_1}\we\cdot\cdot\cdot\we\eps_{j_q}$ À϶§ º¸À̸é ÃæºÐÇѵ¥
ÀÌ °æ¿ì¿¡´Â ºÐ¸íÇÏ´Ù.
\end{proof}

\begin{prop}
$(f\we g)\we h=f\we (g\we h)$.
\end{prop}

\begin{proof}
$\Lambda(V^*)$ÀÇ associativity ·ÎºÎÅÍ ºÐ¸íÇÏ´Ù.
\end{proof}

\section*{ 8. Interior multiplication.}

\begin{defn}
For $l\in End(\Lam(V))$,

$l$ is {\bf derivation} if $l(u\we v)=l(u)\we v+u\we l(v)$.

$l$ is {\bf anti-derivation} if $l(u\we v)=l(u)\we v+(-1)^pu\we
l(v)$, $u\in \Lam^p(V)$.

$l$ is {\bf of deg k} if $l : \Lam^p(V) \rightarrow \Lam^{p+k}(V)
,~ \forall p$.
\end{defn}

{\bf Note.} $l$ is an anti-derivation if and only if


$\dis{l(v_1\we\cdot\cdot\cdot \we
v_r)=\sum_{i=1}^r(-1)^{i+1}v_1\we\cdot\cdot \we
l(v_i)\we\cdot\cdot \we v_r
,\,\,\,v_i\in V,\,\,i=1,\cdot\cdot\cdot, r.}$\\

\begin{proof}
$\Rightarrow)l(v_1\we(v_2\we\cdot\cdot\cdot\we
v_r))=l(v_1)(v_2\we\cdot\cdot\cdot\we v_r)+(-1)v_1\we
l(v_2\we\cdot\cdot\cdot\we v_r)$ ÀÌ°í, ±Í³³¹ýÀ» ¾²¸é µÈ´Ù.

$\Leftarrow)u=v_1\we\cdot\cdot\cdot\we v_p,
v=v_{p+1}\we\cdot\cdot\cdot\we v_{p+q}$ ÀÎ °æ¿ì¸¦ º¸À̸é ÃæºÐÇÏ´Ù.

$l(u\we v)=l(v_1\we\cdot\cdot\cdot\we v_{p+q})$

$\hs{3.5em}=\sum_i^p(-1)^{i+1}v_1\we\cdot\cdot\we l(v_i)\cdot\cdot
v_p\we v_{p+1}\we\cdot\cdot \we v_{p+q})+$

$\hs{4em}(-1)^p\sum_j^q(-1)^{j+1}v_1\we\cdot\cdot\we
v_{p+1}\we\cdot\cdot l(v_{p+j})\cdot\cdot\we v_{p+q}$

$\hs{3.5em}=l(u)\we v+(-1)^pu\we l(v).$

\end{proof}

\begin{defn}
For $x\in V$, define $i(x)=i_x:\Lam(V^*)(\cong A(V))
)\rightarrow\Lam(V^*)(\cong A(V))$ by


$(i_x(f))(v_2,\cdot\cdot\cdot, v_r)=f(x\we
v_2\we\cdot\cdot\cdot\we v_r)$.

\end{defn}

\begin{prop}
$i_x$ is an anti-derivation of deg $-1$.
\end{prop}
\begin{proof}
 À§ Note¸¦ »ç¿ëÇؼ­

$(i_x(\alp^1\we\cdot\cdot\cdot\we\alp^r),v_2\we\cdot\cdot\cdot\we
v_r)=(\sum_{i}^r(-1)^{i+1}\alp^1\we\cdot\cdot\we
i_x(\alp^i)\we\cdot\cdot\we \alp^r,v_2\we\cdot\cdot\cdot\we v_r)$

¸¦ º¸ÀÌ¸é µÈ´Ù. $x=v_1$À̶ó µÎ¸é À§ ½ÄÀÇ  Áº¯Àº
$(\alp^1\we\cdot\cdot\cdot\we\alp^r)(v_1\we
v_2\we\cdot\cdot\cdot\we v_r)$ À̹ǷΠÀÌ´Â $det(\alp(v))$°¡ µÇ°í
¿ìº¯¿¡¼­ $i_x(\alp^i)=\alp^i(v_1)$Àº »ó¼öÀ̹ǷÎ

$\dis{(\sum_{i=1}^r(-1)^{i+1}\alp^1\we\cdot\cdot\we
i_x(\alp^i)\we\cdot\cdot\we \alp^r,v_2\we\cdot\cdot\cdot\we v_r)}$

$\dis{=\sum_{i=1}^r(-1)^{i+1}\alp^i(v_1)(\alp^1\we\cdot\cdot\we
\hat{\alp^i}\we\cdot\cdot \we \alp^r,v_2\we\cdot\cdot\cdot\we
v_r)}$

ÀÌ´Â $det(\alp(v))$ÀÇ  ù¹ø° ¿­º¤ÅÍ¿¡ ´ëÇÑ cofactor
expansionÀ̹ǷΠµû¶ó¼­ Áº¯°ú ¿ìº¯Àº ¸ðµÎ $det(\alp(v))$·Î °°´Ù.


\end{proof}
\end{document}