More generally, the dimension of the subspace of solutions equals the number of free variables: we can think of the free variables as parameters for describing points in the subspace of solutions.

# 18. Geometric viewpoint on simultaneous equations, 2

## 18. Geometric viewpoint on simultaneous equations, 2

### 3 variables

Now let's look at higher dimensional systems.

Consider the equations $$x-y+z=1,x-z=0.$$ The set of points whose coordinates satisfy these equations is a subset of 3-dimensional space ${\mathbf{R}}^{3}$ . Each equation cuts out a plane; their intersection (the set of simultaneous solutions) is a line. :

The solution of the equations is $z=x$ , $y=2x-1$ ($x$ is a free variable); this gives us a parametrisation of the line of intersection: it consists of vectors of the form $\left(\begin{array}{c}\hfill x\hfill \\ \hfill 2x-1\hfill \\ \hfill x\hfill \end{array}\right)$ .

Each equation ${a}_{1}x+{a}_{2}y+{a}_{3}z=b$ gives us a plane. This plane is orthogonal to the vector $\left(\begin{array}{c}\hfill {a}_{1}\hfill \\ \hfill {a}_{2}\hfill \\ \hfill {a}_{3}\hfill \end{array}\right)$ . The $z$ -intercept is where $x=y=0$ , so $z=b/{a}_{3}$ .

Let's consider the system $$x-y+z=1,x-z=0,x-y=0.$$ The first two planes are the same as before; the third equation gives a third plane. The set of solutions is the set of triple intersections between these planes. Solving, we get $x=y=z$ and $z=y=2z-1$ , so $x=y=z=1$ . Therefore there is a unique triple intersection point at $(1,1,1)$ .

### Higher dimensions

An equation ${a}_{1}{x}_{1}+\mathrm{\cdots}+{a}_{n}{x}_{n}=b$
in $n$
variables defines a *hyperplane* in $n$
-dimensional space ${\mathbf{R}}^{n}$
. This is the hyperplane orthogonal to the vector $\left(\begin{array}{c}\hfill {a}_{1}\hfill \\ \hfill \mathrm{\cdots}\hfill \\ \hfill {a}_{n}\hfill \end{array}\right)$
.

If $v=\left(\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {x}_{n}\hfill \end{array}\right)$
is a vector of variables and $Av=b$
is a system of $m$
simultaneous equations encoded with an $m$
-by-$n$
matrix $A$
then each row of $A$
defines a hyperplane ${A}_{i1}{x}_{1}+\mathrm{\cdots}+{A}_{in}{x}_{n}={b}_{i}$
in ${\mathbf{R}}^{n}$
. The set of solutions is a *subspace* of ${\mathbf{R}}^{n}$
. Subspace is a catch-all name, which includes points, lines, planes, hyperplanes, and everything in between (for which we don't have everyday words).

Consider the equations $w+x+y+z=0$ and $x-y=1$ . This is a pair of equations in four variables, so defines a subspace of solutions in ${\mathbf{R}}^{4}$ . Each equation gives a 3-dimensional hyperplane in ${\mathbf{R}}^{4}$ . Their intersection is a 2-dimensional subspace (plane) living in ${\mathbf{R}}^{4}$ . Solving the equations: $w=-2y-z$ , $x=1-y$ . We see that $y,z$ are free variables and $w,x$ are dependent variables. The fact that there are two free variables is another way of saying that the subspace of solutions is 2-dimensional: each free variable is a coordinate on the subspace of solutions.