ABSTRACT
Unfortunately, the word “algebra” carries a rather imprecise meaning from the most elementary and early exposures from which it came to signify the collection of operations done in arithmetic, at the time when the operations are generalized to include symbols or literals such as
a
,
b
, and
c
or
x
,
y
, and
z
. Such a notion generally corresponds closely with the idea of a field,
F
, as defined in Chapter 1, and is not much off the target for an environment of scalars. It may, however, come as a bit of a surprise to the reader that algebra is a technical term, in the same spirit as fields, vector spaces, rings, etc. Therefore, if one is to have available a notion of multiplication of vectors, then it is appropriate to introduce the precise notion of an
algebra
, which captures the desired idea in an axiomatic sense.
