\part{Fundamentals}

\chapter{Linear equations}


Linear algebra originally grew out of the study of simultaneous linear equations. In elementary algebra we introduced the methods of substitution and Gaussian elimination to solve such systems (when a solution exists, that is).

The main insight of linear algebra is modelling simultaneous linear equations by means of linear spaces; an algebraic structure used to model 'multidimensional spaces'. Solving a system of linear equations is analogous to finding the codomain vector of a domain vector of some linear map.


Descartes' analytic geometry was an initial precursor. As the methods of algebra, geometry, and the general formalism of mathematics have advances, the field of linear algebra has come to full fruition. Indeed it is one of the most fundamental, powerful, and versatile tools in the arsenal of a mathemtician, lying at the heart of many mathematic fields relating to geometry and mathematical analysis, mathematical optimization, but also finding applications elsewhere.




Before understanding these methods, it is necessary to understand the nature of linear equations; many calculations in linear algebra will be reduced to these so it is important that we are able to understand the nature of solutions to such problems and how they are calculated.
\section{Gaussian elimination}
Readers are familiar with the idea of simultaneous equations from elementary algebra.  The methods of \emph{substitution or elimination} may be used to find a solution to such a system of equations.



We develop a concrete algorithm to solve linear systems of equations based on the idea of elimination.


%\section{Linear combination}
\subsection{Non-unique solutions}



Though we have been understanding such things from an elementary point of view, the advent of \emph{linear spaces} and \emph{linear maps} will give us the machinery to study non-uniqueness of solutions with more precision and understanding.
