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Vector space consisting of affine subsets
In linear algebra, the quotient of a vector space V {\displaystyle V} by a subspace U {\displaystyle U} is a vector space obtained by "collapsing" U {\displaystyle U} to zero. The space obtained is called a quotient space and is denoted V / U {\displaystyle V/U} (read " V {\displaystyle V} mod U {\displaystyle U} " or " V {\displaystyle V} by U {\displaystyle U} ").
Formally, the construction is as follows.[1] Let V {\displaystyle V} be a vector space over a field K {\displaystyle \mathbb {K} } , and let U {\displaystyle U} be a subspace of V {\displaystyle V} . We define an equivalence relation ∼ {\displaystyle \sim } on V {\displaystyle V} by stating that x ∼ y {\displaystyle x\sim y} iff x − y ∈ U {\displaystyle x-y\in U} . That is, x {\displaystyle x} is related to y {\displaystyle y} if and only if one can be obtained from the other by adding an element of U {\displaystyle U} . This definition implies that any element of U {\displaystyle U} is related to the zero vector; more precisely, all the vectors in U {\displaystyle U} get mapped into the equivalence class of the zero vector.
The equivalence class – or, in this case, the coset – of x {\displaystyle x} is defined as
and is often denoted using the shorthand [ v ] = v + U {\displaystyle [v]=v+U} .
The quotient space V / U {\displaystyle V/U} is then defined as V / ∼ {\displaystyle V/_{\sim }} , the set of all equivalence classes induced by ∼ {\displaystyle \sim } on U {\displaystyle U} . 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 V / U {\displaystyle V/U} into a vector space over K {\displaystyle \mathbb {K} } with U {\displaystyle U} being the zero class, [ 0 ] {\displaystyle [0]} .
The mapping that associates to v ∈ V {\displaystyle v\in V} the equivalence class [ v ] {\displaystyle [v]} is known as the quotient map.
Alternatively phrased, the quotient space V / U {\displaystyle V/U} is the set of all affine subsets of V {\displaystyle V} which are parallel to U {\displaystyle U} [4]
Lines in Cartesian Plane[edit]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.)
Subspaces of Cartesian Space[edit]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.
Polynomial Vector Space[edit]Let P 3 ( R ) {\displaystyle {\mathcal {P}}_{3}(\mathbb {R} )} be the vector space of all cubic polynomials over the real numbers. Then P 3 ( R ) / ⟨ x 2 ⟩ {\displaystyle {\mathcal {P}}_{3}(\mathbb {R} )/\langle x^{2}\rangle } 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 { x 3 + a x 2 − 2 x + 3 : a ∈ R } {\displaystyle \{x^{3}+ax^{2}-2x+3:a\in \mathbb {R} \}} , while another element of the quotient space is { a x 2 + 2.7 x : a ∈ R } {\displaystyle \{ax^{2}+2.7x:a\in \mathbb {R} \}} .
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.[5]
Lebesgue Integrals[edit]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
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).
Isomorphism Theorems[edit] First Isomorphism Theorem[edit]Let V,W be K-Vector Spaces and T:V->W linear. Define the map T ¯ : V / ker T → im ( T ) {\displaystyle {\overline {T}}:V/\ker T\to \operatorname {im} (T)} by T ¯ ( [ v ] ) = T ( v ) . {\displaystyle {\overline {T}}([v])=T(v).} Then T ¯ {\displaystyle {\overline {T}}} is well-defined and an isomorphism.
Quotient of a Banach space by a subspace[edit]If X 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 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.
Generalization to locally convex spaces[edit]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.
If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space, then so is X/M.[9]
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