As discussed in Chapter 1, there are many ways in which you might try to solve a system of linear equation involving a ﬁnite number of variables. These supplementary notes are intended to illustrate the use of Linear Algebra in solving such systems. In particular, any arbitrary number of equations in any number of unknowns — as long as both are ﬁnite —can be encoded as a single matrix equation. As you will see, this has many computational advantages, but, perhaps more importantly, it also allows us to better understand linear systems abstractly. Speciﬁcally, by exploiting the deep connection between matrices and so-called linear maps, one can completely determine all possible solutions to any linear system.
These notes are also intended to provide a self-contained introduction to matrices and important matrix operations. As you read the sections below, remember that a matrix is, in general, nothing more than a rectangular array of real or complex numbers. Matrices are not linear maps. Instead, a matrix can (and will often) be used to deﬁne a linear map.