Vektory b�ze {$ M $} ozna�me po �ad� {$ \vec u_{i},i = 1,2, $} vektory
b�ze {$ N $} ozna�me {$ \vec V_{i}, i=1,2,3. $} Spo�t�me obrazy b�zov�ch
vektor�:
$$
\align
  \vec U_{1}&:=F(\vec u_{1}) = (2,2,2),\\
  \vec U_{2}&:=F(\vec u_{2}) = (0,-2,1).\\
 \endalign
$$
Dal��m krokem  je v�po�et sou�adnic vektor� {$\vec U_{i} $} vzhledem k
b�zi {$ N; $} ozna�me je {$ \alpha _{i},\beta _{i},\gamma _{i},i=1,2.
$} Budeme tedy �e�it syst�m t�� rovnic pro nezn�m�
{$ \alpha ,\beta,\gamma  $} s matic�
$$
\pmatrix
  5      &  -9   &   5 \\
  1      &  -2   &   1 \\
  2      &   3   &   3 \\
\endpmatrix
$$
a s pravou stranou postupn� {$ (2,2,2) \, (i=1), \ (0,-2,1) \ (i=2).$}
Ob� tyto rovnice m��eme �e�it soub�n� ��dkov�mi �pravami matice
roz���en� o ob� prav� strany:

$$
\multline
\left(
  \matrix
  5      &  -9   &   5    \\
  1      &  -2   &   1    \\
  2      &   3   &   3    \\
  \endmatrix
  \right|\!\!\!\left|
  \matrix
 2  & 0\\
 2  &-2\\
 2  & 1\\
 \endmatrix
 \right)
 \sim[III-2\cdot II@>>> III]\sim
\left(
  \matrix
  5      &  -9   &   5    \\
  1      &  -2   &   1    \\
  0      &   7   &   1    \\
  \endmatrix
  \right|\!\!\!\left|
  \matrix
 2  & 0\\
 2  &-2\\
-2  & 5\\
 \endmatrix
 \right)
\sim \\
\sim [I-5\cdot II @>>> I],\
  [II\leftrightarrow I]
\sim\left(
  \matrix
  1      &  -2   &   1    \\
  0      &   1   &   0    \\
  0      &   7   &   1    \\
  \endmatrix
  \right|\!\!\!\left|
  \matrix
 2  &-2\\
-8  &10\\
-2  & 5\\
 \endmatrix
 \right)\sim
 \\
 \sim[-7\cdot II+ III\rightarrow III]
\sim\left(
  \matrix
  1      &  -2   &   1    \\
  0      &   1   &   0    \\
  0      &   0   &   1    \\
  \endmatrix
  \right|\!\!\!\left|
  \matrix
   2  &  -2\\
  -8  &  10\\
  54  &-65\\
 \endmatrix
 \right)\sim
 [  2\cdot II+I@>>> I]
\sim\\
\sim\left(
  \matrix
  1      &   0   &   1    \\
  0      &   1   &   0    \\
  0      &   0   &   1    \\
  \endmatrix
  \right|\!\!\!\left|
  \matrix
 -14  &  18\\
  -8  &  10\\
  54  & -65\\
 \endmatrix
 \right)\sim
 [I- III @>>> I]
\sim\left(
  \matrix
  1      &   0   &   0    \\
  0      &   1   &   0    \\
  0      &   0   &   1    \\
  \endmatrix
  \right|\!\!\!\left|
  \matrix
 -68  &  83\\
  -8  &  10\\
  54  & -65\\
 \endmatrix
 \right);
\endmultline
 $$
 za d�lic� ��rou v posledn�m v�raze vznikla hledan� matice.

