Linear Algebra

Linear algebra is the study of linear transformations on finite vector spaces. $\mathbb{R}^2$ is an example of a vector space, and linear transformations in $\mathbb{R}^n$ are systems of $n$-dimensional linear equations. However, vector spaces and linear transformations are more more general than systems of linear equations. For example, the set of all polynomial functions in $\mathbb{R}$ forms a vector space, and differentiation of polynomials is a linear transformation.

Much of the immediate application of linear algebra comes in the form of solving systems of linear equations, computing geometric values such as volumes and angles, and visualizing vector fields arising from differential equations. These numerical applications make heavy use of matrices, or rectangular grids of numbers. Linear algebra has endless other applications to statistics, multivariable calculus, linear programming, computer graphics, deep learning, supercomputing, and many other fields.

There are two prevailing opinions about how linear algebra ought to be taught. One side argues that most students are interested in the subject from a computational perspective, so pure math proofs and theorems should take a back seat to widely applicable formulas and algorithms. The other side argues that such a computational approach does not provide a proper mathematical understanding of the deeper concepts, so proofs should lead and applications will fall out naturally. Mathmatique takes the proof-centric approach for ordering concepts, but emphasizes applications as soon as and wherever possible.