**An
Example of Generalization in**

**The Keys
to Linear Algebra**

The progression you have just seen from an
ordered list of two numbers to an ordered list of *n*
numbers is an example of a mathematical technique called **generalization**.
Generalization is the process of creating, from an original
concept (problem, definition, theorem, and so on), a more general
concept (problem, definition, theorem, and so on) that includes
not only the original one, but many other new ones as well.}

Each of the original concepts that gives rise
to the generalization is called a **special case**.
In the foregoing examples, the ordered lists

(73, 175) and (73, 175, 25)

are special cases of an *n*-vector **u
**= (*u*_{1}, ..., *u*_{n}).
The first ordered list, (73, 175), is a special case in which *n*
= 2, *u*_{1} = 73, and *u*_{2} =
175, so **u **= (73, 175) e *R*^{2}. The
second ordered list, (73, 175, 25), is a special case in which *n*
= 3, *u*_{1 }= 73, *u*_{2} = 175,
and *u*_{3 }= 25, so **u **= (73,
175, 25) e *R*^{3}. Observe that each of the special
cases is obtained from the generalization by an appropriate *substitution
of values*. Each special case of an *n*-vector **u
**= (*u*_{1}, ..., *u*_{n})
is obtained by substituting specific values for *n* and *u*_{1},
..., *u*_{n}.