## We think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.

Linear algebra is a fundamental discipline underlining anything one can do with Math. From Physics to machine learning, probability theory (ex: Markov chains), you name it. No matter what you’re doing, linear algebra is always lurking under the covers, ready to spring at you as soon as things go multi-dimensional. In my experience (and I’ve heard this from others), this was at the source of a big shock between high school and university. In high school (India), I was exposed to some very basic linear algebra (mainly determinants and matrix multiplication). Then in university level engineering education, every subject all of a sudden seems to be assuming proficiency in concepts like Eigen values, Jacobians, etc. like you were supposed to be born with the knowledge.

This blog is meant to provide a high level overview of the concepts and their obvious applications that exist and are important to know in this discipline. So that you at least know what you don’t know (if anything). Its also an excuse to collect resources and links so people can dig deeper into the rabbit hole.

As mentioned in the previous section, linear algebra inevitably crops up when things go multi-dimensional. We start off with a scalar, which is just a number of some sort. For this article, we’ll be considering real and complex numbers for these scalars. In general, a scalar can be any object where the basic operations of addition, subtraction, multiplication and division are defined (abstracted as a “field”). Now, we want a framework to describe collections of such numbers (add dimensions). These collections are called “vector spaces”. We’ll be considering the cases where the elements of the vector space are either real or complex numbers (the former being a special case of the latter). The resulting vector spaces are called “real vector spaces” and “complex vector spaces” respectively.

The ideas in linear algebra are applicable to these “vector spaces”. The most common example is your floor, table or the computer screen you’re…