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Quotient space (linear algebra)

Index Quotient space (linear algebra)

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. [1]

33 relations: Banach space, Closed set, Codimension, Cokernel, Complete metric space, Coset, Dimension (vector space), Direct sum of modules, Epimorphism, Equivalence class, Equivalence relation, Exact sequence, Field (mathematics), Fréchet space, Hilbert space, Isomorphism, Isomorphism theorems, Kernel (algebra), Kernel (linear algebra), Linear algebra, Linear map, Linear subspace, Locally convex topological vector space, Metrization theorem, Natural transformation, Norm (mathematics), Quotient group, Quotient module, Quotient space (topology), Rank–nullity theorem, Uniform norm, Vector space, Well-defined.

Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.

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Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f: X → Y is the quotient space Y/im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain).

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Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

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Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.

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Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

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Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.

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Epimorphism

In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f: X → Y that is right-cancellative in the sense that, for all morphisms, Epimorphisms are categorical analogues of surjective functions (and in the category of sets the concept corresponds to the surjective functions), but it may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring-epimorphism.

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Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

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Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

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Exact sequence

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Fréchet space

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Isomorphism theorems

In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects.

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Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.

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Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,.

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Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.

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Locally convex topological vector space

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces.

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Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.

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Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.

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Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

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Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

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Quotient module

In algebra, given a module and a submodule, one can construct their quotient module.

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Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

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Rank–nullity theorem

In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix.

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Uniform norm

In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Well-defined

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value.

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Redirects here:

Linear quotient space, Quotient vector space.

References

[1] https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)

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