Linear system/Homogeneous and inhomogeneous/Affine space/1/Example

The solution space of the homogeneous linear equation

${}7x-3y+4z=0\,$

is

${}U=\langle {\begin{pmatrix}3\\7\\0\end{pmatrix}},\,{\begin{pmatrix}0\\4\\3\end{pmatrix}}\rangle \,,$

the solution set of the inhomogeneous linear equation

${}7x-3y+4z=2\,$

is

${}E={\begin{pmatrix}0\\0\\{\frac {1}{2}}\end{pmatrix}}+U={\begin{pmatrix}0\\0\\{\frac {1}{2}}\end{pmatrix}}+\langle {\begin{pmatrix}3\\7\\0\end{pmatrix}},\,{\begin{pmatrix}0\\4\\3\end{pmatrix}}\rangle \,.$

The affine addition is the mapping

$U\times E\longrightarrow E,(u,P)\longmapsto (u+P),$

which assigns to a pair consisting in a solution of the homogeneous equation and a solution of the inhomogeneous equation their sum, which is a solution of the inhomogeneous equation. For two solutions of the inhomogeneous equation, their difference is a solution of the homogeneous equation. For example, for

${}u=-2{\begin{pmatrix}3\\7\\0\end{pmatrix}}+3{\begin{pmatrix}0\\4\\3\end{pmatrix}}={\begin{pmatrix}-6\\-2\\9\end{pmatrix}}\in U\,,$

the point

${}{\begin{pmatrix}0\\0\\{\frac {1}{2}}\end{pmatrix}}+{\begin{pmatrix}-6\\-2\\9\end{pmatrix}}={\begin{pmatrix}-6\\2\\{\frac {19}{2}}\end{pmatrix}}\,$

is another solution in ${}E$. The two solutions ${}{\begin{pmatrix}1\\3\\1\end{pmatrix}}$ and ${}{\begin{pmatrix}2\\4\\0\end{pmatrix}}$ from ${}E$ are related by the translating vector

${}{\begin{pmatrix}2\\4\\0\end{pmatrix}}-{\begin{pmatrix}1\\3\\1\end{pmatrix}}={\begin{pmatrix}1\\1\\-1\end{pmatrix}}\,.$