### Contravariance and Covariance - Part 6

##### Summary
The idea of contravariance and covariance is more an algebraic concept than a geometric one, and it expresses the relation between vectors, dual vectors, and the invariance of scalars under a transformation. I could have easily omitted parts 3 and 4 on tensor analysis and manifolds, and went straight to category theory, since the relevant definitions in 3 and 4 are motivated by and may be thought of as an extension of the algebraic concepts found in parts 1 and 2.
The distinction between contravariance and covariance may even appear in a rectangular coordinate system (I alluded to this in part 1), though it is easy to find texts that mistakenly state otherwise. Therefore, no differential geometry, curved spaces, or even calculus, is required. You do however have to know how to do a coordinate transformation. This is why I started my discussion with oblique coordinates and the reciprocal lattice in part 1.
(By the way, if it is not clear that contravariance and covariance can appear even in rectangular coordinates, take a look at section 1.3 of "Methods of Theoretical Physics", part 1 by Morse and Feshbach, and in particular, pages 30-31 on contravariant and covariant vectors, where they show it is sufficient to merely introduce a scale factor in the new coordinate system.)
##### Further Study
This coverage of contravariance/covariance has been brief, so where to go for more? Ideally, a University class (of course). If you live near a University that even offers these subjects, consider yourself fortunate. In this era of perpetually increasing University tuition costs, you might think finding a course in Topology, or advanced Linear Algebra, or Differential Geometry would be easy. What you tend to find however is the opposite. Usually the student interest in these courses is limited, and enrollments are small. Departments may switch to offering them alternate years, if at all.
If you opt for self-study, access to good books is essential. Being able to persuse the library stacks at your local University even if they don't offer the course is a great help. Study via the internet is a little problematical, since, as Anders Ericsson says, you can find anything on the internet but quality control.
Should you find yourself in the situation of having to learn these ideas on your own, I would recommend you start first with Linear Algebra. Mastery (*) of Linear Algebra will immediately set you apart from most of your peers and open a wide range of subjects to you. For example, Dirac's formulation of Quantum Mechanics allows you to study it as an application of Linear Algebra, rather than Partial Differential Equations (i.e.,the Schroedinger Equation) And if you have any interest in Quantum Computing, you will need it. Linear Algebra will also provide a segue into Category Theory.

(*) These subjects are advanced and self-study does always introduce the possibility of missing key ideas, making mastery more difficult. Here for example is Rob Pike of Golang fame, who I shall assume is self-taught in Linear Algebra, at 9:44, not comprehending what properties a vector space has, nor what a basis is.
I have previously given the definition of a vector space in this series, part 2. There is NO notion of orthogonal. You can add an inner product to the vector space, turning it to an inner product space at which point you can in fact talk about orthogonality. So Pike's statement that a basis set is orthogonal is, in general, incorrect.
Pike's use of cover "a vector basis covering a solution space" is meaningless in a vector space. In a vector space, one refers to the span of a vector(s). The notion of cover does not exist in a vector space. The notion of cover does exist in a Topological space. Interestingly enough, in a Topological Space, there is the notion of a base that generates the topology. Perhaps he heard cover used in the Topological sense, and did not understand the difference. I can only speculate.