**An Example of
Abstraction in**

**The Keys to Linear
Algebra**

The key to unifying different types of items,
such as *n*-vectors and matrices, is to use the following
mathematical technique. **Abstraction** is the
process of taking the focus farther and farther away from
specific items by working with general *objects*. Thus,
you become more abstract---hence the term "abstract
mathematics.''

To illustrate the idea of abstraction in a non-mathematical setting, consider apples and oranges. You can unify these two items into the single comprehensive class of fruits (fruits include apples and oranges as special cases). You can then apply generalization by considering, instead of fruits, the more general class of foods (foods include fruits as a special case). With abstraction, you broaden the class even further by considering objects rather than specific items like foods or fruits. By thinking of objects, you can now include in the same group such diverse items as foods, computers, houses, and much more.

Turning to *n*-vectors and matrices, you
can use abstraction to include these two diverse items in a
single group by thinking of objects rather than *n*-vectors
and matrices. That is, you can create a set of objects, say, *V*.
The elements of *V* can all be *n*-vectors,
matrices, or other items. Thus, the set *V *contains both *n*-vectors
and matrices as special cases.

Abstraction allows you to unify *n*-vectors
and matrices, but one of the disadvantages of doing so is that
you lose the properties of the specific items that give rise to
the abstraction. For example, you know how to add two *n*-vectors
**u** and **v**, however, you cannot
"add'' two objects **u** and **v **from
an arbitrary set *V*, so a syntax error results when you
write

u+v

You will now see how to overcome this problem.