In linear algebra, the quotient of a vector space
is a vector space obtained by "collapsing"
get mapped into the equivalence class of the zero vector.
The equivalence class – or, in this case, the coset – of
, the set of all equivalence classes induced by
Scalar multiplication and addition are defined on the equivalence classes by[2][3] It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representatives).
These operations turn the quotient space
[4] Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X.
That is to say that, the elements of the set X/Y are lines in X parallel to Y.
Note that the points along any one such line will satisfy the equivalence relation because their difference vectors belong to Y.
(By re-parameterising these lines, the quotient space can more conventionally be represented as the space 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.)
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 (x1, ..., xn).
The subspace, identified with Rm, consists of all n-tuples such that the last n − m entries are zero: (x1, ..., xm, 0, 0, ..., 0).
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.
be the vector space of all cubic polynomials over the real numbers.
is a quotient space, where each element is the set corresponding to polynomials that differ by a quadratic term only.
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:[6][7] Let T : V → W be a linear operator.
The kernel of T, denoted ker(T), is the set of all x in V such that Tx = 0.
The kernel is a subspace of V. The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).
The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).
We define a norm on X/M by Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm.
Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R. If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M. The quotient of a locally convex space by a closed subspace is again locally convex.
[8] Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα | α ∈ A} where A is an index set.
Let M be a closed subspace, and define seminorms qα on X/M by Then X/M is a locally convex space, and the topology on it is the quotient topology.