# Linear Algebra/Linear Geometry of n-Space

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|previous=Automation
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.

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*Unique solution*

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*No solutions*

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*Infinitely many*\
*solutions*

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![](Linalg_unique_solution.png "Linalg_unique_solution.png"){width="150"}\
$\begin{array}{*{2}{rc}r}
\scriptstyle 3x  &\scriptstyle +  &\scriptstyle 2y  &\scriptstyle =  &\scriptstyle 7   \\[-5pt]
\scriptstyle x   &\scriptstyle -  &\scriptstyle y   &\scriptstyle =  &\scriptstyle -1
\end{array}$

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![](Linalg_no_solutions.png "Linalg_no_solutions.png"){width="150"}\
$\begin{array}{*{2}{rc}r}
\scriptstyle 3x  &\scriptstyle +  &\scriptstyle 2y  &\scriptstyle =  &\scriptstyle 7   \\[-5pt]
\scriptstyle 3x  &\scriptstyle +  &\scriptstyle 2y  &\scriptstyle =  &\scriptstyle 4
\end{array}$

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![](Linalg_infinitely_many_solutions.png "Linalg_infinitely_many_solutions.png"){width="150"}\
$\scriptstyle \begin{array}{*{2}{rc}r}
\scriptstyle 3x  &\scriptstyle +  &\scriptstyle 2y  &\scriptstyle =  &\scriptstyle 7   \\[-5pt]
\scriptstyle 6x  &\scriptstyle +  &\scriptstyle 4y  &\scriptstyle =  &\scriptstyle 14
\end{array}$

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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.

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