Linear Algebra: A second course, featuring proofs and Python

At an elementary level, linear algebra is the study of vectors (in ${\mathbb{R}}^n$) and matrices. Of course, much of that study revolves around systems of equations. Recall that if $\mathbf{x}$ is a vector in ${\mathbb{R}}^n$ (viewed as an $n\times 1$ column matrix), and $A$ is an $m\times n$ matrix, then $\mathbf{y}=A\mathbf{x}$ is a vector in ${\mathbb{R}}^m\text{.}$ Thus, multiplication by $A$ produces a function from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m\text{.}$

This example motivates the definition of a linear transformation, and as we’ll see, provides the archetype for all linear transformations in the finite-dimensional setting. Many areas of mathematics can be viewed at some fundamental level as the study of sets with certain properties, and the functions between them. Linear algebra is no different. The sets in this context are, of course, vector spaces. Since we care about the linear algebraic structure of vector spaces, it should come as no surprise that we’re most interested in functions that preserve this structure. That is precisely the idea behind linear transformations.