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In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N). Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...
Definition
Formally, the construction is as follows. Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. Let [x] denote the equivalence class containing x. We define scalar multiplication and addition on the equivalent classes by Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...
In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...
- α[x] = [αx] for all α ∈ K, and
- [x] + [y] = [x+y].
It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). These operations turn the quotient space V/N into a vector space over K. In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...
Examples and properties This simplest example is to take a quotient of Rn. Let m ≤ n and let Rm be the subspace spanned by the first m standard basis vectors. Two vectors of Rn are then seen to be equivalent 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. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
In general, if V is n-dimensional vector space and U is an m-dimensional subspace, the quotient space V/U has dimension n−m.
Let T : V → W be a linear operator. The kernel (or nullspace) of T, denoted ker(T) is the set of all x ∈ V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem of linear algebra 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 that 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). In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
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. ...
The null space (also nullspace) of a matrix A is the set of all vectors v which solve the equation Av = 0, a linear subspace of the space of all vectors. ...
In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
Quotient of a Banach space by a subspace 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 In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
The quotient space X/M is complete with respect to the norm, so it is a Banach space. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
Examples Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1]. 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. In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
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