Representation of an associative algebra

Okay kiddo, let's start with what an algebra is. An algebra is like a set of rules we use to do math problems with different things like numbers, variables, or even matrices.

An associative algebra is a special type of algebra, where we can multiply things together in any order, and it doesn't matter which order we choose.

Now, when we talk about a representation of an associative algebra, it's like finding a way to show the algebra using something else that we understand better.

It's like if we have a bunch of Legos, and we want to make a house out of them, but we don't know how to make the house. We can ask someone else to show us how to make the house by giving us a set of instructions.

This is what a representation is: it's a set of instructions that shows how to make a house out of Legos, but instead of Legos, we're showing how to use the rules of the algebra.

So basically, a representation is a way to "translate" the rules of the algebra into a different language, so that we can understand it better and work with it more easily.