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\begin{document}

\title{�㐔�w���_II�i�x�N�g����Ԃ̘a�Ɛρj}

\author{�R�c�q�G(Tomohiro Yamada)}
\date{}
\maketitle

$U, V$ �� $\BK$ ��̃x�N�g����Ԃł���Ƃ�
$U\cap V$ �� $\BK$ ��̃x�N�g����Ԃł���A $U, V$ �̕�����Ԃł���B
���� $\Bu, \Bv\in U, V$ �Ȃ�� $\Bu+\Bv, k\Bu\in U, V$ �ł���B
�Ⴆ��3������ԓ��̕��ʂƕ��ʂ̌����͕��ʂ܂��͒����ł���B

����P���Șa�W�� $U\cup V$ �̓x�N�g����ԂƂȂ�Ƃ͌���Ȃ��B
�Ⴆ��2�{�̕��s�ł͂Ȃ������̘a�W���̓x�N�g����Ԃł͂Ȃ��B���
$\Bu+\Bv (\Bu\in U, \Bv\in V)$ �̌`�̃x�N�g���S�̂� $\BK$ ��̃x�N�g����ԂƂȂ�B
���� $\Bu_1+\Bv_1+\Bu_2+\Bv_2=(\Bu_1+\Bu_2)+(\Bv_1+\Bv_2)$
$k(\Bu+\Bv)=k\Bu+k\Bv$ �ł���B������
$\Bu+\Bv (\Bu\in U, \Bv\in V)$ �̌`�̃x�N�g���S�̂̂Ȃ���Ԃ�
$U, V$ ��{\bf �a���}�Ƃ��� $U+V$ �ł���킷�B

$U=\BK\langle \Bu_1, \Bu_2, \ldots, \Bu_m\rangle, V=\BK\langle \Bv_1, \Bv_2, \ldots, \Bv_n\rangle$
�ł���Ƃ� $U+V=\BK\langle \Bu_1, \Bu_2, \ldots, \Bu_m, \Bv_1, \Bv_2, \ldots, \Bv_n\rangle$
�ł���B�܂� $U+V$ �� $U$ �̃x�N�g���� $V$ �̃x�N�g���Ő���������Ԃł���Ƃ�������B

������ $\Bu_1, \Bu_2, \ldots, \Bu_m$ �� $U$ �̊���
$\Bv_1, \Bv_2, \ldots, \Bv_n$ �� $V$ �̊��ł����Ă�
$\Bu_1, \Bu_2, \ldots, \Bu_m, \Bv_1, \Bv_2, \ldots, \Bv_n$ �� $U+V$ �̊��ł���Ƃ�
����Ȃ��B
\begin{example}
$\Bu_1=\vecxyzw 1021, \Bu_2=\vecxyzw 1201$
$\Bv_1=\vecxyzw 1111, \Bv_2=1100$ �̂Ƃ�
$\Bu_1, \Bu_2$ �� $U=\{2x=2w=y+z\}$ �̊��A
$\Bv_1, \Bv_2$ �� $V=\{x=y, z=w\}$ �̊��ł���B
������ $\Bu_1+\Bu_2-2\Bv_1=\Bzr$ ������ $\Bu_1, \Bu_2, \Bv_1, \Bv_2$ �͐��`�Ɨ��ł͂Ȃ��B
���� $U+V=\{x+w=y+z\}$ �ł��� $\dim U+V=3$ �ł���B
\end{example}

\paragraph{}

�ȑO�A$\Bv_1, \Bv_2, \ldots, \Bv_n$ �����`�Ɨ��� $m<n$ �Ȃ��
$m$ �̃x�N�g�� $\Bw_1, \Bw_2, \ldots, \Bw_m$ �͂�͂� $\BK\langle\Bv_1, \Bv_2, \ldots, \Bv_n\rangle$ ��
��ꂽ�肦�Ȃ��Ə������B

����ɐi��Ŏ��̂��Ƃ��킩��B

\begin{lem}
$\Bv_1, \Bv_2, \ldots, \Bv_n$ �����`�Ɨ���
$m<n$ �̂Ƃ��x�N�g����� $V=\BK\langle\Bv_1, \Bv_2, \ldots, \Bv_n\rangle$ ����
$m$ �̃x�N�g�� $\Bw_1, \Bw_2, \ldots, \Bw_m$ �ɑ΂�
$\BK\langle\Bv_1, \Bv_2, \ldots, \Bv_n\rangle$ ���̃x�N�g�� $\Bu$ ��
$\BK\langle\Bw_1, \Bw_2, \ldots, \Bw_m\rangle$ �Ɋ܂܂�Ȃ����̂����݂���B
����ɂ��̂Ƃ� $\Bw_1, \Bw_2, \ldots, \Bw_m$ �����`�Ɨ��Ȃ�΁A����� $\Bu$ ��������
$\Bw_1, \Bw_2, \ldots, \Bw_m, \Bu$ �����`�Ɨ��ł���B
\end{lem}
\begin{proof}
�ȑO�ɂ��q�ׂ��Ƃ��� $U=\BK\langle\Bw_1, \Bw_2, \ldots, \Bw_m\rangle$ ��
$V=\BK\langle\Bv_1, \Bv_2, \ldots, \Bv_n\rangle$ ���܂ނȂ��
�Ȃ�� $n>m$ ��� $\Bv_1, \Bv_2, \ldots, \Bv_n$ �͐��`�]���łȂ���΂Ȃ�Ȃ��B
����� $V$ �� $U$ �ɂ͊܂܂�Ȃ��̂� $V$ �̒��� $U$ ��
�܂܂�Ȃ��x�N�g�� $\Bu$ �����݂���B
�܂� $k_1\Bw_1+k_2\Bw_2+\cdots +k_m\Bw_m+k_{m+1}\Bu=\Bzr$ �Ȃ��
$k_{m+1}\Bu=-(k_1\Bw_1+k_2\Bw_2+\cdots +k_m\Bw_m)\in U$
�ł��邪 $\Bu$ $U$ �ɂ͊܂܂�Ȃ��̂� $k_{m+1}=0$ ���Ȃ킿
$k_1\Bw_1+k_2\Bw_2+\cdots +k_m\Bw_m=\Bzr$ �ł���B
$\Bw_1, \Bw_2, \ldots, \Bw_m$ �����`�Ɨ��Ȃ�� $k_1=k_2=\cdots =k_m=0$ �Ȃ̂�
$k_1=k_2=\cdots =k_m=k_{m+1}=0$ �ƂȂ�B�����$\Bw_1, \Bw_2, \ldots, \Bw_m, \Bu$ ��
���`�Ɨ��ł��邱�Ƃ��Ӗ����Ă���B
\end{proof}

���� $U$ �� $n$ �����x�N�g����ԁA $V$ �����̕�����ԂŊ�ꂪ $\Bv_1, \Bv_2, \ldots, \Bv_m (m\leq n)$
�ŗ^������Ƃ��ȉ��̂悤�ɂ��� $U$ �̊��� $\Bv_1, \Bv_2, \ldots, \Bv_m$ ���܂ނ��̂��Ƃ��:

$m=n$ �̂Ƃ��� $\Bv_1, \Bv_2, \ldots, \Bv_m$ �����̂܂� $U$ �̊��ƂȂ�B
$m<n$ �̂Ƃ��� $U$ �ɑ����邪 $V$ �ɂ͑����Ȃ��x�N�g�� $\Bw_1$ �����݂���B���̂Ƃ�
��̕���� $\Bv_1, \Bv_2, \ldots, \Bv_m, \Bw_1$ �͐��`�Ɨ��ł��邩��
\begin{equation*}
V_1=\BK\langle\Bv_1, \Bv_2, \ldots, \Bv_m, \Bw_1\rangle
\end{equation*}
�Ƃ����� $V_1$ �� $U$ �� $m+1$ �����̕�����Ԃł���B
$m+1=n$ �Ȃ�� $V_1=U$ �ł���B
$m+1<n$ �Ȃ�� $U$ �ɑ����邪 $V_1$ �ɂ͑����Ȃ��x�N�g�� $\Bw_2$ �����݂���B���̂Ƃ�
$\Bv_1, \Bv_2, \ldots, \Bv_m, \Bw_1, \Bw_2$ �͐��`�Ɨ��ł���B
\begin{equation*}
V_2=\BK\langle\Bv_1, \Bv_2, \ldots, \Bv_m, \Bw_1, \Bw_2\rangle
\end{equation*}
�Ƃ����� $V_2$ �� $U$ �� $m+2$ ������
������Ԃł���B������J��Ԃ������� $n$ �ɒB����܂ŌJ��Ԃ���
\begin{equation*}
V_{n-m}=\BK\langle\Bv_1, \Bv_2, \ldots, \Bv_m, \Bw_1, \ldots, \Bw_{n-m}\rangle
\end{equation*}
 �� $U$ �Ɉ�v����B


\begin{lem}
$U\cap V$ �̊��� $\Bx_1, \Bx_2, \ldots, \Bx_l$ �Ƃ���B
���̂Ƃ�
$\By_1, \By_2, \ldots, \By_m, \Bx_1, \Bx_2, \ldots, \Bx_l$ �� $U$ �̊���
$\Bz_1, \Bz_2, \ldots, \Bz_n, \Bx_1, \Bx_2, \ldots, \Bx_l$ �� $V$ �̊��Ȃ��

$\Bx_1, \ldots, \Bx_l, \By_1, \ldots, \By_m, \Bz_1, \ldots, \Bz_n$ �� $U+V$ �̊��ł���B
\end{lem}
\begin{proof}
$\Bu\in U, \Bv\in V$ �ɑ΂���
\begin{equation*}
\begin{split}
\Bu=& a_1\Bx_1+\cdots +a_l\Bx_l+a_{l+1}\By_1+\cdots +a_{l+m}\By_m, \\
\Bv=& b_1\Bx_1+\cdots +b_l\Bx_l+b_{l+1}\Bz_1+\cdots +b_{l+n}\Bz_n
\end{split}
\end{equation*}
�Ƃ�����
\begin{equation*}
\begin{split}
\Bu+\Bv=& (a_1+b_1)\Bx_1+\cdots +(a_l+b_l)\Bx_l \\
& +a_{l+1}\By_1+\cdots +a_{l+m}\By_m \\
& +b_{l+1}\Bz_1+\cdots +b_{l+n}\Bz_n
\end{split}
\end{equation*}
�ł��邩��
\begin{equation*}
U+V=\BK\langle\Bx_1, \ldots, \Bx_l, \By_1, \ldots, \By_m, \Bz_1, \ldots, \Bz_n\rangle
\end{equation*}
�ł��邱�Ƃ͂����ɂ킩��B
���� $\Bx_1, \ldots, \Bx_l, \By_1, \ldots, \By_m, \Bz_1, \ldots, \Bz_n$ �����`�Ɨ��ł��邱�Ƃ������B
���̂��߂�
\begin{equation*}
\begin{split}
\Bzr= & a_1\Bx_1 +\cdots +a_l\Bx_l \\
+ & b_1\By_1 +\cdots +b_m\By_m \\
+ & c_1\Bz_1 +\cdots +c_n\Bz_n
\end{split}
\end{equation*}
 �Ƃ���B
\begin{equation*}
\begin{split}
\Bp=& a_1\Bx_1 +\cdots +a_l\Bx_l, \\
\Bq=& b_1\By_1 +\cdots b_m\By_m, \\
\Br=& c_1\Bz_1 +\cdots +c_n\Bz_n
\end{split}
\end{equation*}
�Ƃ����� $\Bp+\Bq+\Br=0$ �ł���B
������ $\Br=-(\Bp+\Bq)$ �ł��邪 $\Br\in V, \Bp, \Bq\in U$ �ł��邱�Ƃ���
$\Br\in U\cap V$ �ł���B������ $\Bx_1, \Bx_2, \ldots, \Bx_l$ �� $U\cap V$ �̊��ł��邱�Ƃ���
\begin{equation*}
\Br=c_1\Bz_1 +\cdots +c_n\Bz_n=d_1\Bx_1+d_2\Bx_2+\cdots +d_l\Bx_l
\end{equation*}
�܂�
\begin{equation*}
c_1\Bz_1 +\cdots +c_n\Bz_n-(d_1\Bx_1+d_2\Bx_2+\cdots +d_l\Bx_l)=\Bzr
\end{equation*}
�ƂȂ邪 $\Bz_1, \Bz_2, \ldots, \Bz_n, \Bx_1, \Bx_2, \ldots, \Bx_l$ �͐��`�Ɨ��Ȃ̂�����
$\Br=\Bzr$ �łȂ���΂Ȃ�Ȃ��B�Ƃ��낪�A����� $\Bz_1, \ldots, \Bz_n$ �����`�Ɨ��ł��邩��
���� $c_1=c_2=\cdots =c_n=0$ �łȂ���΂Ȃ�Ȃ��B

��� $a_1\Bx_1 +\cdots +a_l\Bx_l +b_1\By_1 +\cdots b_m\By_m=\Bp+\Bq=-\Br$ ���܂���x�N�g���ł��邪
$\By_1, \By_2, \ldots, \By_m, \Bx_1, \Bx_2, \ldots, \Bx_l$ �͐��`�Ɨ��ł��邩��
$a_1=\cdots =a_l=b_1=\cdots =b_m=0$ �łȂ���΂Ȃ�Ȃ��B���Ȃ킿
$\Bx_1, \ldots, \Bx_l, \By_1, \ldots, \By_m, \Bz_1, \ldots, \Bz_n$ �͐��`�Ɨ��ł���B
\end{proof}

��̕����A���̂悤�Ȋ��͎��ۂɑ��݂���̂�
$\dim(U\cap V)=l, \dim U=m+l, \dim V=n+l$ �Ƃ�����
$\dim(U+V)=m+n+l=(m+l)+(n+l)-l$ �ł���B����Ď�������
\begin{equation}
\dim(U\cap V)+\dim(U+V)=\dim U+\dim V
\end{equation}
�����藧�B

���� $U\cap V=\{\Bzr\}$ �ł���Ƃ�
\begin{equation*}
\dim U+\dim V=\dim (U+V)
\end{equation*}
�����藧�B���ۂ��̂Ƃ� $\Bu_1, \Bu_2, \ldots, \Bu_m$ �� $U$ �̊���
$\Bv_1, \Bv_2, \ldots, \Bv_n$ �� $V$ �̊��Ȃ��
$\Bu_1, \Bu_2, \ldots, \Bu_m, \Bv_1, \Bv_2, \ldots, \Bv_n$ �� $U+V$ �̊��ƂȂ�B
�܂��t��
\begin{equation*}
\dim U+\dim V=\dim (U+V)
\end{equation*}
�Ȃ�� $\dim(U\cap V)=0$ ���
$U\cap V=\{\Bzr\}$ �ł��邱�Ƃ��킩��B
�����ł��̂悤�ȏꍇ�A���� $U+V$ �� $U$ �� $V$ ��{\bf ���a}�Ƃ���
$U\oplus V$ �Ƃ���킷�B

\begin{example}
$M(n; \BK)$ �� $\BK$ ��� $n$ �������s��S�̂̏W���A
$S(n; \BK)$ �� $^t A=A$ �ƂȂ� $\BK$ ��� $n$ �������s��S�̂̏W���A
$A(n; \BK)$ �� $^t A=-A$ �ƂȂ� $\BK$ ��� $n$ �������s��S�̂̏W���Ƃ����
$S(n; \BK), A(n; \BK)$ �͋��� $M(n; \BK)$ �̕�����Ԃ�
$M(n; \BK)=S(n; \BK)\oplus A(n; \BK)$ �ł���B
\end{example}

\begin{example}
$\Bu_1=\vecxyz{1}{2}{-2 \\ 1}, \Bu_2=\vecxyz{2}{3}{-1 \\ -1},
\Bv_1=\vecxyz{-1}{0}{-3 \\ 2}, \Bv_2=\vecxyz{-2}{-1}{-3 \\ 1}$ �Ƃ�
$U=\BK\langle\Bu_1, \Bu_2\rangle, V=\BK\langle\Bv_1, \Bv_2\rangle$ �Ƃ���B

���̂Ƃ�
\begin{equation*}
\begin{split}
& \left(\begin{array}{cccc|cccc}
  1 &  2 & -1 & -2 & 0 \\
  2 &  3 &  0 & -1 & 0 \\
 -2 & -1 & -3 & -3 & 0 \\
  1 & -1 &  2 &  1 & 0
\end{array}\right)\\
\rightarrow & 
\left(\begin{array}{cccc|cccc}
  1 &  2 & -1 & -2 & 0 \\
  0 & -1 &  2 &  3 & 0 \\
  0 &  3 & -1 &  1 & 0 \\
  0 & -3 &  3 &  3 & 0
\end{array}\right)\\
\rightarrow & 
\left(\begin{array}{cccc|cccc}
  1 &  2 & -1 & -2 & 0 \\
  0 & -1 &  2 &  3 & 0 \\
  0 &  0 &  5 & 10 & 0 \\
  0 &  0 & -3 & -6 & 0
\end{array}\right)\\
\rightarrow & 
\left(\begin{array}{cccc|cccc}
  1 &  2 & -1 & -2 & 0 \\
  0 & -1 &  2 &  3 & 0 \\
  0 &  0 &  5 & 10 & 0 \\
  0 &  0 &  0 &  0 & 0 \\
\end{array}\right)\\
\end{split}
\end{equation*}
��� $a\Bu_1+b\Bu_2+c\Bv_1+d\Bv_2=\Bzr$ �̉���
$c=-2d, b=2c+3d=-d, a=-2b+c+2d=2d$ �܂� $(a, b, c, d)=(2d, -d, -2d, d)$ �ŗ^������B
����� $U\cap V$ �� $\Bw=2\Bu_1-\Bu_2=2\Bv_1-\Bv_2$ �Ő�������邱�Ƃ��킩��B
�܂� $\Bu_1, \Bw$ �� $U$ �̊��, $\Bv_1, \Bw$ �� $V$ �̊���
$U+V$ �̊��� $\Bu_1, \Bv_1, \Bw$ �ŗ^������B
\end{example}


\noindent
\end{document}
