What is an Algebra? (Part III)

In our next few Algebras, we broaden our sense of elements. Until now, we’ve looked at different types of numbers as our elements.

Now, consider matrices as elements. Given any two 2×2 matrices, and , we can certainly computer and . And crucially, this sum and this product are still 2×2 matrices. So, we see that the set of 2×2 matrices forms an Algebra. It shouldn’t be too surprising that the set of 3×3 matrices also forms an Algebra, and more generally the set of x matrices do as well.

We could also use functions (say, ones that turn a real number into another real number) as our elements. In that case, we have addition and multiplication of functions as operations. And we have another reasonable operation to look into–composition.

Do we have division in the world of matrices or the world of functions? If you multiply a matrix A by a matrix B, is there another matrix to multiply by that will get you back to A? What about something that undoes composition by or multiplication by in the same way that subtracting undoes adding ?