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# 07 Geometric interpretation of simple linear systems -- hyperplanes.
When the number of variables is small, say $\le 3$, then we can graph out each linear equation and see what is happening.
Case $n=2$ variables.
If we just have two variables, say $x$ and $y$, then a linear equation $$
ax+by=c
$$ is just a **line** in $2$-space.
So solving a system of linear equations in $x$ and $y$ is just finding what are all the points that are simultaneously on all of the lines $$
\left\{
\begin{array}{}
a_{1} x + b_{1}y & = c_{1} \\
a_{2} x + b_{2}y & = c_{2} \\
\vdots \\
a_{k} x + b_{k}y & = c_{k}
\end{array}\right.
$$
And one can imagine all the situations that could happen. Either all the lines intersect at the same point (to which we get unique solution); all the lines are on top of each other (infinitely many solutions, over $\mathbb{R}$); or some lines are parallel or three or more lines that do not all meet at the same points (inconsistent system, no solution).
For example here are a system of two equations in two variables $x$ and $y$ with unique solutions. Each equation is a line.
```tikz
\begin{document}
\begin{tikzpicture}[scale=1.5]
% Coordinate axes
\draw[->] (-1,0) -- (3,0) node[right] {\(x\)};
\draw[->] (0,-0.5) -- (0,2.5) node[above] {\(y\)};
% Red line
\draw[red, solid] (-0.5,-0.5) -- (2.5,2.5) node[right] {};
% Blue line
\draw[blue, solid] (-0.5,1.5) -- (2.5,-0.5) node[right] {};
% Intersection point
\filldraw[black] (0.7,0.7) circle (1pt) node[anchor=south west] {};
\end{tikzpicture}
\end{document}
```
Here depicts a system with two equation in two variables $x$ and $y$ with infinitely many solutions. Each equation is a line and they are on top of each other.
```tikz
\begin{document}
\begin{tikzpicture}[scale=1.5]
% Coordinate axes
\draw[->] (-1,0) -- (3,0) node[right] {\(x\)};
\draw[->] (0,-0.5) -- (0,2.5) node[above] {\(y\)};
% Red line
\draw[red, dashed, thick] (-0.5,0.51) -- (2.5,1.51) node[right] {};
% Blue line
\draw[blue, solid] (-0.5,0.5) -- (2.5,1.5) node[right] {};
\end{tikzpicture}
\end{document}
```
And here depicts a system of two equations in two variables $x$ and $y$ with no solutions. These are parallel:
```tikz
\begin{document}
\begin{tikzpicture}[scale=1.5]
% Coordinate axes
\draw[->] (-1,0) -- (3,0) node[right] {\(x\)};
\draw[->] (0,-0.5) -- (0,2.5) node[above] {\(y\)};
% Red line
\draw[red, solid] (-0.5,0.5) -- (2.5,1.5) node[right] {};
% Blue line
\draw[blue, solid] (-0.5,1.5) -- (2.5,2.5) node[right] {};
\end{tikzpicture}
\end{document}
```
Another situation with no solutions: Three equations in two variables $x,y$ where the lines do not have a single common intersection to all of them:
```tikz
\begin{document}
\begin{tikzpicture}[scale=1.5]
% Coordinate axes
\draw[->] (-1,0) -- (3,0) node[right] {\(x\)};
\draw[->] (0,-0.5) -- (0,2.5) node[above] {\(y\)};
% Red line
\draw[red, solid] (-0.5,0.5) -- (2.5,1.5) node[right] {};
% Blue line
\draw[blue, solid] (-0.5,1.3) -- (2.5,0.3) node[right] {};
% Green line
\draw[olive, solid] (-0.5,2.5) -- (2.5,-0.5) node[right] {};
\end{tikzpicture}
\end{document}
```
Case $n=3$ variables.
If we have three variables, say $x,y,z$, then we can plot each equation in $3$-space $\mathbb{R^{3}}$, assuming it is over the real numbers $\mathbb{R}$. If one "recalls" from multivariable geometry, each linear equation $$ax+by+cz=d$$ describes a **plane** in $3$-space.
Below we depict how one can have unique solution, multiple (infinite) solutions, and no solutions with linear equations in three variables $x,y,z$.
Here are three equations in three variables $x,y,z$ with unique solution:
```tikz
\begin{document} \begin{tikzpicture}[scale=1.5]
% Define coordinates
\coordinate (O) at (0,0,0);
% Draw axes
\draw[->] (O) -- (2,0,0) node[right] {\(y\)};
\draw[->] (O) -- (0,2,0) node[above] {\(z\)};
\draw[->] (O) -- (0,0,2) node[below left] {\(x\)};
% Draw planes
\filldraw[blue!30, opacity=0.4] (0,0,1) -- (2,0,1) -- (2,2,1) -- (0,2,1) -- cycle;
% XY plane
\filldraw[green!30, opacity=0.4] (0,1,0) -- (2,1,0) -- (2,1,2) -- (0,1,2) -- cycle;
% XZ plane
\filldraw[red!30, opacity=0.4] (1,0,0) -- (1,2,0) -- (1,2,2) -- (1,0,2) -- cycle;
% YZ plane
% Draw intersection point
\draw [dashed, gray] (0,1,1) -- (2,1,1);
\draw [dashed, gray] (1,0,1) -- (1,2,1);
\draw [dashed, gray] (1,1,0) -- (1,1,2);
\filldraw[black] (1,1,1) circle (1pt) node[above right] {Intersection Point};
\end{tikzpicture}
\end{document}
```
Here we have infinitely many solutions, where their common intersection forms a line:
```tikz
\begin{document} \begin{tikzpicture}[scale=1.5]
% Define coordinates
\coordinate (O) at (0,0,0);
% Draw axes
\draw[->] (O) -- (2,0,0) node[right] {\(y\)};
\draw[->] (O) -- (0,2,0) node[above] {\(z\)};
\draw[->] (O) -- (0,0,2) node[below left] {\(x\)};
% Draw planes
\filldraw[blue!30, opacity=0.4] (0,0,1) -- (2,0,1) -- (2,2,1) -- (0,2,1) -- cycle;
% XY plane
\filldraw[green!30, opacity=0.4] (0,1,0) -- (2,1,0) -- (2,1,2) -- (0,1,2) -- cycle;
% XZ plane
\filldraw[red!30, opacity=0.4] (0,0,0) -- (0,2,2) -- (2,2,2) -- (2,0,0) -- cycle;
% YZ plane
% Draw intersection point
\draw [dashed] (0,1,1) -- (2,1,1) node[above right] {Intersection Line};
\end{tikzpicture}
\end{document}
```
And here we have no solution, where there is no point common to every equation:
```tikz
\begin{document} \begin{tikzpicture}[scale=1.5]
% Define coordinates
\coordinate (O) at (0,0,0);
% Draw axes
\draw[->] (O) -- (2,0,0) node[right] {\(y\)};
\draw[->] (O) -- (0,2,0) node[above] {\(z\)};
\draw[->] (O) -- (0,0,2) node[below left] {\(x\)};
% Draw planes
\filldraw[blue!30, opacity=0.4] (0,0,1.5) -- (2,0,1.5) -- (2,2,1.5) -- (0,2,1.5) -- cycle;
% XY plane
\filldraw[green!30, opacity=0.4] (0,1,0) -- (2,1,0) -- (2,1,2) -- (0,1,2) -- cycle;
% XZ plane
\filldraw[red!30, opacity=0.4] (0,0,0) -- (0,2,2) -- (2,2,2) -- (2,0,0) -- cycle;
% YZ plane
\end{tikzpicture}
\end{document}
```
**Remark.** In higher dimensions it is harder to envision these objects, but in general you should keep this in mind: A linear equation $$
c_{1}x_{1}+ c_{2}x_{2}+\cdots +c_{n}x_{n}=d,
$$linear in variables $x_{1},x_{2},\ldots,x_{n}$ with $c_{1},c_{2},\ldots,c_{n},d$ are scalars, is just a **flat object** in higher dimensional space -- it's just a flat thingy. When we have 2 variables, it is a line; when we have 3 variables, it is a plane. The proper terminology is a **hyperplane**. That is, the equation $c_{1}x_{1}+c_{2}x_{2}+\cdots +c_{n}x_{n}=d$ describes an $(n-1)$-dimensional hyperplane in $n$-space, over whatever field the scalars $c_{1},c_{2},\ldots,c_{n},d$ lives in.