IA Vectors and Matrices
We might have first learned vectors as arrays of numbers, and then defined
addition and multiplication in terms of the individual numbers in the vector.
This however, is not what we are going to do here. The array of numbers is just
a representation of the vector, instead of the vector itself.
Here, we will define vectors in terms of what they are, and then the various
operations are defined axiomatically according to their properties.