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 = (u1, ..., un). The first ordered list, (73, 175), is a special case in which n = 2, u1 = 73, and u2 = 175, so u = (73, 175) e R2. The second ordered list, (73, 175, 25), is a special case in which n = 3, u1 = 73, u2 = 175, and u3 = 25, so u = (73, 175, 25) e R3. 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 = (u1, ..., un) is obtained by substituting specific values for n and u1, ..., un.