Abstract linear spaces are also called vector spaces, and they occur in a wide variety of mathematical settings. The most familiar examples are the finite dimensional Euclidean spaces. Let N be a positive integer. Then ℜ^N can be represented as the set of all column vectors

in which each coordinate x_n, 1nN is an element from the field ℜ of real numbers. Sometimes it is more convenient to represent the elements of ℜ^N as row vectors, but for the purposes of this article column vectors will be used. There are two basic algebraic operations that are defined in ℜ^N. The first is addition of vectors. If x and y are two vectors in ℜ^N then x + y is the vector obtained by adding the corresponding coordinate of x and y. Thus if we write the vectors as columns...