# 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. The space obtained is called a **quotient space** and is denoted *V*/*N* (read *V* mod *N*).

## Contents

## 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*. The equivalence class of *x* is often denoted

- [
*x*] =*x*+*N*

since it is given by

- [
*x*] = {*x*+*n*:*n*∈*N*}.

The quotient space *V*/*N* is then defined as *V*/~, the set of all equivalence classes over *V* by ~. Scalar multiplication and addition are defined on the equivalence classes by

- α[
*x*] = [α*x*] 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*.

## Examples and properties

This simplest example is to take a quotient of **R**^{n}. Let *m* ≤ *n* and let **R**^{m} be the subspace spanned by the first *m* standard basis vectors. Two vectors of **R**^{n} are then seen to be equivalent if and only if they are identical in the last *n*−*m* coordinates. The quotient space **R**^{n}/ **R**^{m} is isomorphic to **R**^{n−m} in an obvious manner.

More generally, if *V* is written as an (internal) direct sum of subspaces *U* and *W*:

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then the quotient space *V*/*U* is naturally isomorphic to *W*.

If *U* is a subspace of *V*, the **codimension** of *U* in *V* is defined to be the dimension of *V*/*U*. If *V* is finite-dimensional, this is just the difference in the dimensions of *V* and *U*:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U).}**

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\to U\to V\to V/U\to 0.\,}**

Let *T* : *V* → *W* be a linear operator. The kernel 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 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*).

## 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X. }**

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

### 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*.