from F such that only finitely many elements are nonzero. That is, if we write an element of F∞ as then only a finite number of the xi are nonzero (i.e., the coordinates become all zero after a certain point). Addition and scalar multiplication are given as in finite coordinate space. The dimensionality of F∞ is countably infinite. A standard basis consists of the vectors ei which contain a 1 in the i-th slot and zeros elsewhere. This vector space is the coproduct (or direct sum) of countably many copies of the vector space F. Note the role of the finiteness condition here. One could consider arbitrary sequences of elements in F, which also constitute a vector space with the same operations, often denoted by FN - see below. FN is the product of countably many copies of F. By Zorn's lemma, FN has a basis (there is no obvious basis). There are uncountably infinite elements in the basis. Since the dimensions are different, FN is not isomorphic to F∞. It is worth noting that FN is (isomorphic to) the dual space of F∞, because a linear map T from F∞ to F is determined uniquely by its values T(ei) on the basis elements of F∞, and these values can be arbitrary. Thus one sees that a vector space need not be isomorphic to its double dual if it is infinite dimensional, in contrast to the finite dimensional case. Product of vector spaces Starting from n vector spaces, or a countably infinite collection of them, each with the same field, we can define the product space like above. Matrices Let Fm×n denote the set of m×n matrices with entries in F. Then Fm×n is a vector space over F. Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar). The zero vector is just the zero matrix. The dimension of Fm×n is mn. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0. When m = n the matrix is square and matrix multiplication of two such matrices produces a third. This vector space of dimension n2 forms an algebra over a field. Polynomial vector spaces One variable The set of polynomials with coefficients in F is a vector space over F, denoted F[x]. Vector addition and