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Home > Quotient space (linear algebra)

1 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

• 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.

2 Examples and properties

This simplest example is to take a quotient of Rⁿ. Let m ≤ n and let R^m 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/ R^m is isomorphic to R^n−m in an obvious manner.

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 immediately corollary (for finite-dimensional spaces) is that the dimension of V is equal to the dimenison of the kernel (the nullity of T) plus the dimension of the image (the rank of T).

3 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

The quotient space X/M is complete with respect to the norm, so it is a Banach space.

3.1 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.

4 Related topics

• quotient set
• quotient groupIn mathematics, given a group G and a normal subgroup N of G the quotient group or factor group of G over N is a group that intuitively "collapses" the normal subgroup N to the identity element. The quotient group is written G ''N and is usually spoken in
• quotient moduleIf B is a submodule of a module A of a ring R, then the quotient space A/B is also a module of R. It's called the quotient module .
• quotient spaceFor quotient spaces in linear algebra, see quotient space (linear algebra). 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" certai (in topologyTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g)

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Topics: Quotient Space (linear Algebra) Subspace Vector Banach X''/''m...

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Quotient groupIn mathematics, given a group G and a normal subgroup N of G the quotient group or factor group of G over N is a group that intuitively "collapses" the normal subgroup N to the identity element. The quotient group is written G ''N and is usually spoken in Quotient fieldEvery integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. The elements of the quotient field of the integral domain R have the form a/b with a and b in R Quotient ruleCalculus In calculus, the quotient rule is a method of finding the derivative of a function which is the quotient of two other functions for which derivatives exist. If the function one wishes to differentiate, f ''x , can be written as : and h ''x ≠ 0 Quotient spaceFor quotient spaces in linear algebra, see quotient space (linear algebra). 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" certai QuotientIn mathematics, a quotient is the end result of a division problem. For example, in the problem 6 ÷ 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor. A quotient can also mean just the integral part of the result of divid Quotient moduleIf B is a submodule of a module A of a ring R, then the quotient space A/B is also a module of R. It's called the quotient module . Quotient space (linear algebra)