Inlinear algebra, the quotient of a vector space by a subspace
is a vector space obtained by "collapsing"
to zero. The space obtained is called a quotient space and is denoted
(read "
mod
" or "
by
").
Formally, the construction is as follows.[1] Let be a vector space over a field
, and let
be a subspaceof
. We define an equivalence relation
on
by stating that
if
. That is,
is related to
if one can be obtained from the other by adding an element of
. From this definition, one can deduce that any element of
is related to the zero vector; more precisely, all the vectors in
get mapped into the equivalence class of the zero vector.
The equivalence class – or, in this case, the coset – of is often denoted
since it is given by
The quotient space is then defined as
, the set of all equivalence classes induced by
on
. 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 into a vector space over
with
being the zero class,
.
The mapping that associates to the equivalence class
is known as the quotient map.
Alternatively phrased, the quotient space is the set of all affine subsetsof
which are parallelto
.[4]
Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. 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. This gives a way to visualize quotient spaces geometrically. (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/RmisisomorphictoRn−m in an obvious manner.
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
.
More generally, if V is an (internal) direct sum of subspaces U and W,
then the quotient space V/Uisnaturally isomorphictoW.[5]
An important example of a functional quotient space is an Lp space.
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
IfU is a subspace of V, the dimensionofV/U is called the codimensionofUinV. Since a basisofV may be constructed from a basis AofU and a basis BofV/U by adding a representative of each element of BtoA, the dimension of V is the sum of the dimensions of U and V/U. If Visfinite-dimensional, it follows that the codimension of UinV 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 xinV 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 imageofVinW. 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 nullityofT) plus the dimension of the image (the rankofT).
The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).
IfX is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a normonX/Mby
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.
IfX is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complementofM.
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 topologyonX 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 qαonX/Mby
Then X/M is a locally convex space, and the topology on it is the quotient topology.
If, furthermore, Xismetrizable, then so is X/M. If X is a Fréchet space, then so is X/M.[9]
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Basic concepts |
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Matrices |
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Bilinear |
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Multilinear algebra |
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Vector space constructions |
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Numerical |
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