Linear Algebra/Linear Geometry of n-Space

For readers who have seen the elements of vectors before, in calculus or physics, this section is an optional review. However, later work will refer to this material so it is not optional if it is not a review.

In the first section, we had to do a bit of work to show that there are only three types of solution sets— singleton, empty, and infinite. But in the special case of systems with two equations and two unknowns this is easy to see. Draw each two-unknowns equation as a line in the plane and then the two lines could have a unique intersection, be parallel, or be the same line.

Unique solution	No solutions	Infinitely many solutions
${\begin{array}{*{2}{rc}r}\scriptstyle 3x&\scriptstyle +&\scriptstyle 2y&\scriptstyle =&\scriptstyle 7\\[-5pt]\scriptstyle x&\scriptstyle -&\scriptstyle y&\scriptstyle =&\scriptstyle -1\end{array}}$	${\begin{array}{*{2}{rc}r}\scriptstyle 3x&\scriptstyle +&\scriptstyle 2y&\scriptstyle =&\scriptstyle 7\\[-5pt]\scriptstyle 3x&\scriptstyle +&\scriptstyle 2y&\scriptstyle =&\scriptstyle 4\end{array}}$	$\scriptstyle {\begin{array}{*{2}{rc}r}\scriptstyle 3x&\scriptstyle +&\scriptstyle 2y&\scriptstyle =&\scriptstyle 7\\[-5pt]\scriptstyle 6x&\scriptstyle +&\scriptstyle 4y&\scriptstyle =&\scriptstyle 14\end{array}}$

These pictures don't prove the results from the prior section, which apply to any number of linear equations and any number of unknowns, but nonetheless they do help us to understand those results. This section develops the ideas that we need to express our results from the prior section, and from some future sections, geometrically. In particular, while the two-dimensional case is familiar, to extend to systems with more than two unknowns we shall need some higher-dimensional geometry.