A linear space, also called a vector space, is a set with well-defined properties. First, a linear space must be defined over a field. This field is often the field of real numbers or the field of complex numbers, but may be other fields as well. The field elements are known as scalars, and the elements of the linear space are known as vectors. (These vectors should not be confused with physical vectors of two- or three-dimensions--although those vectors are elements of linear spaces.) This space needs to have defined at least two operations, addition and scalar multiplication, that fit the following criteria:

• Addition must be commutative. This means that for all pairs x,y in a linear space L, x + y = y + x.
• Addition must also be associative. For all x,y,z in L, (x + y) + z = x + (y + z).
• Zero exists. There must...