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鐃緒申鐃緒申鐃緒申鐃緒申鐃� I No.14

\fbox{試験解答例}

鐃緒申鐃緒申 14.1   鐃緒申鐃縮に￥申鐃縦器申鐃緒申 $R$ 鐃緒申 $R$-鐃獣件申 $M$ 鐃夙わ申鐃出わ申鐃緒申 $\underbrace{M\oplus M\oplus \dots \oplus M}_{\text{$n$\space 個}}$ 鐃舜わ申 $A=(a_{ij})\in M_n(R)$ 鐃塾「縁申鐃緒申鐃術」わ申

\begin{displaymath}(m_1,\dots,m_n).A= (\sum_{j=1}^n a_{j1}.m_j,\ \sum_{j=1}^n a_{j2}.m_j, \dots,\ \sum_{j=1}^n a_{jn}m_{j}) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦�(鐃純え鐃緒申 $n=3$ 鐃淑わ申鐃�

\begin{displaymath}(u,v,w). \begin{pmatrix} a & b & c\\ d & e & f \\ g & h & i \end{pmatrix}= (a.u+d.v+g.w,b.u+e.v+h.w,c.u+f.v+i.w) \tag{★} \end{displaymath}

鐃夙わ申鐃緒申鐃緒申鐃叔わ申鐃暑。) 鐃緒申鐃塾とわ申鐃緒申

\begin{displaymath}(((m_1,\dots,m_n).A).B)=(m_1,\dots,m_n).(AB) \end{displaymath}

鐃緒申任鐃春わ申 $m_1,\dots,m_n \in M$ 鐃緒申鐃緒申鐃� 任鐃春わ申 $A,B\in M_n(R)$ 鐃緒申 鐃出わ申鐃緒申鐃緒申鐃緒申立鐃縦わ申鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申

(鐃緒申鐃緒申): $(m_1,m_2,\dots,m_n)$ 鐃塾わ申鐃夙わ申略鐃緒申鐃緒申$(m_i)$ 鐃夙わ申鐃緒申鐃緒申 $i$ 鐃緒申 $1$ 鐃緒申鐃緒申 $n$ 鐃殉わ申動鐃緒申鐃緒申辰討鐃緒申鐃緒け鐃叔わ申鐃暑が鐃緒申 鐃緒申鐃塾よう鐃緒申略鐃緒申鐃緒申鐃熟緒申鐃緒申鐃熟随申鐃緒申鐃淑学英語ゼ鐃淳ナ￥申鐃緒申任鐃緒申覆鐃緒申澆覆呂鐃緒申任鐃緒申襦� 鐃淑駕申鐃塾わ申鐃曙ぞ鐃緒申亮鐃緒申任匹鐃緒申竸鐃緒申鐃銃逸申鐃緒申鐃暑か鐃祝はとわ申鐃緒申鐃緒申鐃春わ申鐃竣わ申鐃暑。 鐃緒申鐃緒申任鐃淑�鐃緒申鐃緒申砲鐃緒申鐃緒申隼廚鐃緒申佑鐃� $(m_1,m_2,\dots,m_n)$ 鐃緒申 $...$ 鐃緒申箸鐃緒申鐃緒申鐃緒申鐃俊わ申鐃銃わ申茲わ申鐃緒申鐃緒申鐃緒申覆鐃緒申忙鐃緒申砲覆襪鰹申箸鐃� 鐃出悟せ鐃粛ばなわ申覆鐃緒申鐃� 鐃渋の居申鐃緒申 $\sum$ 鐃緒申鐃緒申鐃殉わ申鐃夙わ申鐃暑こ鐃夙￥申 鐃夙わ申鐃祝どわ申添鐃緒申鐃緒申鐃春わ申鐃緒申鐃渋わ申箸辰討鐃緒申襪�鐃緒申鐃緒申鐃緒申如鐃� 鐃緒申鐃緒申鐃緒申鐃緒申任呂鐃緒申鐃緒申鐃緒申鐃緒申屬離櫂鐃緒申鐃夙でわ申鐃暑。 (鐃夙わ申鐃緒申鐃緒申鐃緒申鐃緒申奮鐃緒申鐃緒申鐃緒申鐃熟なわ申鐃緒申鐃緒申) (鐃緒申鐃緒申):

$A=(a_{ij}),B=(b_{kl})$ 鐃夙わ申鐃緒申函鐃�


\begin{alignat*}{2} &((m_i).A).B &\qquad &\leftarrow\text{$i$\space が$1,2,\dot... ...\right) &\qquad& \leftarrow\text{行列の積の定義}\\ =& (m_i).(AB) \end{alignat*}

鐃緒申鐃緒申 14.2   $R=\mbox{${\Bbb Q}$ }[X]$, $V=\mbox{${\Bbb Q}$ }^3 $ ( $\mbox{${\Bbb Q}$ }$ 鐃緒申鐃� $3$ 鐃緒申鐃緒申鐃縦ベワ申鐃夙ワ申鐃緒申鐃�)鐃夙わ申鐃緒申鐃緒申$V$ 鐃緒申 $R$-鐃獣件申鐃塾刻申造鐃緒申

\begin{displaymath}p(X).v=p(A).v \qquad (p\in \mbox{${\Bbb Q}$}[X],\quad v \in V) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申法鐃�

\begin{displaymath}A=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 &2 \end{pmatrix}\tag{※} \end{displaymath}

鐃叔わ申鐃暑。

\begin{displaymath}C=\begin{pmatrix} 1 & 0 & 0\\ X -1 & 1 & -1\\ -X(X-1)(X-2) & - X(X-2) & (X -1)^2\\ \end{pmatrix}\end{displaymath}

,

\begin{displaymath}D= \begin{pmatrix} 0 & 1 & X -2\\ -1 & X -1 & \left(X -1\right) \left(X -2\right)\\ 0 & X & (X-1)^2\\ \end{pmatrix}\end{displaymath}

鐃夙わ申鐃緒申鐃夙￥申

\begin{displaymath}C(XE-A)D= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & (X-1)^2(X-2) \end{pmatrix}\end{displaymath}

鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃塾常申鐃緒申鐃塾駕申鐃叔￥申

1.
鐃緒申鐃緒申14.1鐃塾計誌申鐃緒申(鐃緒申)鐃祝わ申鐃緒申鐃緒申鐃獣わ申

\begin{displaymath}(e_1,e_2,e_3).(XE-A) \end{displaymath}

( $E \in M_3(R)$ 鐃緒申単鐃縮刻申鐃緒申) 鐃緒申彁鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�
2.
$C$ 鐃塾逆刻申鐃緒申鐃竣誌申鐃緒申鐃淑わ申鐃緒申鐃緒申
3.
$v_1,v_2,v_3\in V$ 鐃緒申 $(v_1,v_2,v_3)=(e_1,e_2,e_3).C^{-1}$ 鐃緒申鐃緒申鐃緒申鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申$A$ 鐃塾駈申鐃緒申的鐃淑件申(鐃緒申)鐃緒申鐃所せ鐃緒申鐃緒申鐃緒申$C$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃夙っわ申 $e_1,e_2,e_3$ 鐃緒申 $v_1,v_2,v_3$ 鐃緒申 $R$-鐃緒申鐃緒申鐃塾一次鐃緒申鐃叔書きわ申鐃緒申錣件申覆鐃緒申鐃緒申鐃�

4.
$v_1,v_2,v_3$ 鐃緒申 $\mbox{${\Bbb Q}$ }^3$ 鐃塾醐申鐃夙わ申鐃銃実際わ申鐃緒申分鐃緒申表鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申鐃緒申呂鐃緒申鐃獣とわ申笋鰹申鐃緒申鐃緒申辰鐃緒申鐃緒申鐃緒申里鐃淑わ申鐃緒申 鐃銃ンソワ申鐃術わ申鐃薯けわ申鐃緒申鐃潤、$X$ 鐃緒申鐃緒申弔鐃緒申鐃緒申鐃緒申鐃緒申鐃緒せ鐃銃わ申鐃緒申箸鐃緒申鐃緒申鐃� 鐃緒申鐃緒申蕕件申鐃緒申鐃� 鐃夙わ申鐃祝￥申

\begin{displaymath}X.e_1=\begin{pmatrix} X\\ 0\\ 0 \end{pmatrix}\qquad\text{(まちがい)} \end{displaymath}

鐃緒申鐃塾間違い鐃薯してわ申鐃緒申圓鐃緒申鐃緒申鐃渋随申鐃緒申辰鐃緒申鐃� 鐃緒申鐃曙が鐃盾し鐃緒申鐃緒申 $A$ 鐃緒申鐃出わ申鐃緒申

\begin{displaymath}Ae_1=\begin{pmatrix} A\\ 0\\ 0 \end{pmatrix}\qquad\text{(まちがい)} \end{displaymath}

鐃夙わ申鐃銃わ申鐃緒申里鐃緒申辰鐃緒申鐃緒申鐃緒申鐃盾す鐃緒申鐃祝間違い鐃祝居申鐃重わ申鐃緒申鐃緒申鐃夙と思わ申鐃緒申 $\mbox{${\Bbb Q}$ }^3$ 鐃緒申 $\mbox{${\Bbb Q}$ }[X]$-鐃獣件申鐃夙醐申鐃暑、鐃夙わ申鐃緒申鐃叔緒申鐃淑醐申呂鐃淳わ申鐃夙わ申鐃緒申 $X$ 鐃緒申 $\mbox{${\Bbb Q}$ }^3$ 鐃舜の削申鐃術わ申鐃緒申鐃祝なわ申茲�鐃祝なっわ申頂鐃緒申鐃緒申鐃緒申鐃緒申

(鐃緒申鐃緒申):

1.


\begin{alignat*}{2} &(e_1,e_2,e_3)(XE-A)\\ =&(e_1,e_2,e_3) \begin{pmatrix} X-1 ... ...0 &X-2 \end{pmatrix}\\ =&((X-1).e_1, -e_1+(X-1).e_2, (X-2).e_3) \end{alignat*}
鐃緒申鐃緒申鐃緒申鐃初が鐃緒申鐃緒申如鐃� 鐃緒申鐃緒申砲鐃緒申鐃処う鐃緒申 $X$ 鐃緒申 $e_1,e_2,e_3$ 鐃舜の削申鐃術わ申 $A$ 鐃祝わ申襪�鐃緒申鐃緒申鐃叔わ申鐃暑。 鐃緒申鐃緒申鐃緒申鐃獣て￥申

\begin{displaymath}((A-E)e_1,-e_1+(A-E)e_2, (A-2E)e_3) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃竣誌申鐃緒申鐃緒申弌鐃�$(0,0,0)$ (-鐃緒申鐃緒申)鐃緒申鐃緒申鐃暑。

2. 鐃緒申鐃緒申鐃獣縁申鐃竣誌申鐃叔わ申鐃暑。鐃緒申鐃瞬誌申鐃瞬わ申箸鐃夙わ申鐃緒申鐃叔わ申鐃暑が鐃緒申 鐃緒申鐃緒申鐃塾とわ申鐃緒申鐃緒申鐃緒申鐃緒申率鐃緒申鐃盾か鐃獣わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申

\begin{displaymath}\begin{pmatrix} 1 & 0 & 0\\ - X + 1 & - 2 X + X^2 + 1 & 1\\ 0 & X^2- 2 X & 1 \end{pmatrix}\end{displaymath}

鐃叔わ申鐃暑。 MuPAD 鐃叔わ申鐃出逸申発鐃叔わ申鐃暑が鐃緒申 鐃緒申鐃緒申魯鐃緒申鐃峻ュー鐃緒申鐃緒申鐃緒申鐃塾緒申鐃緒申鐃熟わ申鐃淑わ申鐃獣わ申鐃処う鐃叔わ申鐃暑。

鐃淑わ申鐃緒申鐃緒申鐃緒申鐃塾駕申鐃緒申鐃緒申鐃緒申如鐃�$X$ 鐃緒申鐃術随申鐃叔はなわ申鐃緒申 $\mbox{${\Bbb Q}$ }$ 鐃塾醐申鐃緒申鐃塾わ申鐃夙わ申 鐃緒申鐃縦わ申鐃獣てわ申鐃緒申物鐃緒申鐃緒申鐃獣わ申鐃緒申鐃緒申鐃緒申鐃緒申呂鐃緒申里泙泙任牢岼磴わ申任鐃緒申襦� (鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申法鐃盾あ鐃暑が鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申呂鐃緒申里茲�鐃淑テワ申鐃祝ッワ申鐃緒申箸鐃緒申曚匹鐃� 鐃緒申里任呂覆鐃緒申鐃緒申鐃� 鐃緒申鐃塾駕申鐃緒申鐃緒申颪い鐃緒申佑鐃緒申鐃緒申離謄鐃緒申縫奪鐃緒申鐃緒申箸鐃緒申討鐃緒申鐃夙は考わ申鐃祝わ申鐃緒申鐃緒申)

3. 鐃緒申鐃緒申鐃緒申鐃緒申牢鐃獣縁申鐃緒申鐃緒申鐃緒申里能鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申砲鐃緒申鐃緒申辰鐃緒申里鐃緒申鐃緒申里鐃淑わ申鐃緒申 $(v_1,v_2,v_3)$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申蕁�

\begin{displaymath}(e_1,e_2,e_3)=(v_1,v_2,v_3).C \end{displaymath}

鐃夙わ申鐃緒申鐃舜件申鐃緒申鐃緒申鐃銃易わ申分鐃緒申鐃所、 鐃緒申鐃�
\begin{align*}e_1&=v_1+(X-1).v_2-X(X-1)(X-2).v_3, \\ e_2&=v_2-X(X-2).v_3,\\ e_3&=-v_2+(X-1)^2.v_3. \end{align*}
鐃夙わ申鐃緒申鐃緒申鐃夙になる。(-鐃緒申鐃緒申)

4. 鐃緒申鐃緒申鐃� 2. 鐃塾件申未鐃夙わ申鐃緒申
\begin{align*}v_1&=e_1+(-X+1).e_2, \\ v_2&=(X-1)^2.v_2+X(X-2).e_3,\\ v_3&=e2+e3. \end{align*}
鐃叔わ申鐃所、1. 鐃緒申同鐃粛に計誌申鐃緒申鐃緒申弌鐃�


\begin{displaymath}v_1= \begin{pmatrix}0\\ 0\\ 0\end{pmatrix},\qquad v_2= \begin... ...\end{pmatrix},\qquad v_3= \begin{pmatrix}0\\ 1\\ 1\end{pmatrix}\end{displaymath}

鐃緒申鐃緒申鐃暑。(-鐃緒申鐃緒申)

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2001-01-30