A common fallacy about mathematics is it is very rigid and requires no imagination. 

While the mechanics of an algorithm may exhibit those characteristics, the true nature of mathematical study demands insight into abstract ideas and a willingness to abandon any number of "truths" to spur new areas of exploration.

Calculus is grounded in the concept of limits.  Sometimes limits approach infinity, which is not a number and something that cannot be graphed or pictured easily and which requires a huge leap into the land of imagination.  In Linear Algebra, our intrepid band of scholars recently discovered just about anything is fair game to redefine or change, even the most basic operations of addition and scalar multiplication!

We began the latest chapter by broadly defining an abstraction called a vector space.  The advantage of defining such an abstraction is it can be attached to very disparate objects while still maintaining predictable characteristics.  Additionally, when the spaces share the same dimension, they are at some level equivalent regardless of how they are concretely represented.  The class learned there is something inherently equal about the set of all four-dimensional vectors, square two-by-two matrices, and polynomials of degree three or less.

In other disciplines such as writing and music, it is easy to be creative quickly with even a small skill set.  It takes a lot longer to develop enough skills in mathematics to make that creative leap, but it is surely there.  Mathematical imagination takes a back seat to no subject!

This is part 3 of an ongoing series from Leon "Mr. Calculus" Galitsky. Check out part 1 and part 2!