The primary example of a vector space is Rⁿ, the set of ordered n-tuples of real numbers. If v = (v₁,...,v_n) w =(w₁,..,w_n) are elements of Rⁿ then v + w is defined to be (v₁ + w₁,...,v_n + w_n). If r is a real number, then rv is defined to be (rv₁, rv₂,..., rv_n). The set of real numbers is the field of scalars for Rⁿ. In general, a vector space V over a field K (called the field of scalars) is a set with an operation and an 'action of K'. The operation is denoted by +. It defines vector addition. The 'action of K' defines how an element of K can be multiplied with an element of V. The following rules must also be satisfied for any elements x, y, z in V and k in K...