Representation

In math, representation is the craft of using math objects to take the place of things we find in reality.

We are going to use that $550,000 sports car your girlfriend bought you to travel from home to Cedar Point Amusement Park. We represent that fabulous car by this really simple vector and do a calculation.

Don’t feel bad, somebody represented our country’s trillion-dollar spaceship with a point mass and momentum vector. We argue that these math representations are the simplest way of representing that which must be represented.

If you work with representations, you will soon realize something might be represented by two different ways and both ways are helpful. For example, we might study the movements of shapes and then we devise symbols to represent the movements. We might then say that doing two motions one after the other is multiplication of a sort…

When working with the hours on a clock:

(+4) then (+4) = (-4)

and we might then make a multiplication table to show all possible combinations of two motions which give you a third motion.

Later we find that we can represent these motions by matrices and we can build a multiplication table that takes two matrices multiplied together which gives us a third matrix.

It is meaningful if we show there is a correspondence between these two tables and that is another topic–see Homomorphism.