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 𝐑3 . Each equation cuts out a plane; their intersection (the set of simultaneous solutions) is a line. :

The two planes x-y+z=1 and x-z=0 in 3D

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 (x2x-1x) .

Each equation a1x+a2y+a3z=b gives us a plane. This plane is orthogonal to the vector (a1a2a3) . The z -intercept is where x=y=0 , so z=b/a3 .

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 a1x1++anxn=b in n variables defines a hyperplane in n -dimensional space 𝐑n . This is the hyperplane orthogonal to the vector (a1an) .

If v=(x1xn) 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 Ai1x1++Ainxn=bi in 𝐑n . The set of solutions is a subspace of 𝐑n . 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 𝐑4 . Each equation gives a 3-dimensional hyperplane in 𝐑4 . Their intersection is a 2-dimensional subspace (plane) living in 𝐑4 . 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.

Remark:

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.