of all points along a line through the origin that is not parallel to Y. Similarly, the quotient space for R3 by a line through the origin can again be represented as the set of all co-parallel lines, or alternatively be represented as the vector space consisting of a plane which only intersects the line at the origin.) Subspaces of Cartesian Space Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers . The subspace, identified with Rm, consists of all n-tuples such that the last n − m entries are zero: . Two vectors of Rn are in the same equivalence class modulo the subspace if and only if they are identical in the last n − m coordinates. The quotient space Rn/Rm is isomorphic to Rn−m in an obvious manner. Polynomial Vector Space Let be the vector space of all cubic polynomials over the real numbers. Then is a quotient space, where each element is the set corresponding to polynomials that differ by a quadratic term only. For example, one element of the quotient space is , while another element of the quotient space is . General Subspaces More generally, if V is an (internal) direct sum of subspaces U and W, then the quotient space V/U is naturally isomorphic to W. Lebesgue Integrals An important example of a functional quotient space is an Lp space. Properties There is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may be constructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, the dimension of V is the sum of the dimensions of U and V/U. If V is finite-dimensional, it follows that the codimension of U in V is the difference between the dimensions of V and U: Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x in V