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 $N$" or "$V$ by $N$").

Definition

Formally, the construction is as follows. Let $V$ be a vector space over a field $\mathbb$, and let $N$ be a subspace of $V$. We define an equivalence relation $\sim$ on $V$ by stating that $x \sim y$ iff . That is, $x$ is related to $y$ if and only if one can be obtained from the other by adding an element of $N$. This definition implies that any element of $N$ is related to the zero vector; more precisely, all the vectors in $N$ get mapped into the equivalence class of the zero vector.

The equivalence class – or, in this case, the coset – of $x$ is defined as
: := \
and is often denoted using the shorthand = x + N.

The quotient space $V/N$ is then defined as $V/_\sim$, the set of all equivalence classes induced by $\sim$ on $V$. Scalar multiplication and addition are defined on the equivalence classes by
*\alpha = alpha x/math> for all $\alpha \in \mathbb$, and
* + = +y/math>.
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 $V/N$ into a vector space over $\mathbb$ with $N$ being the zero class, /math>.

The mapping that associates to the equivalence class /math> is known as the quotient map.

Alternatively phrased, the quotient space $V/N$ is the set of all affine subsets of $V$ which are parallel to

Examples

Lines in Cartesian Plane

Let 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

Another example is the quotient of R^''n'' by the subspace spanned by the first ''m'' standard basis vectors. The space R^''n'' consists of all ''n''-tuples of real numbers . The subspace, identified with R^''m'', consists of all ''n''-tuples such that the last ''n'' − ''m'' entries are zero: . Two vectors of R^''n'' 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 R^''n''/R^''m'' is isomorphic to R^{''n''−''m''} in an obvious manner.

Polynomial Vector Space

Let $\mathcal_3(\mathbb)$ be the vector space of all cubic polynomials over the real numbers. Then $\mathcal_3(\mathbb) / \langle x^2 \rangle$ 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 $\$, while another element of the quotient space is $\$.

General Subspaces

More generally, if ''V'' is an (internal) direct sum of subspaces ''U'' and ''W,''
:$V=U\oplus W$
then the quotient space ''V''/''U'' is naturally isomorphic to ''W''.

Lebesgue Integrals

An important example of a functional quotient space is an L^''p'' space.

Properties

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
:$0\to U\to V\to V/U\to 0.\,$

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'':
:$\mathrm(U) = \dim(V/U) = \dim(V) - \dim(U).$

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'').

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
: \, \, _ = \inf_ \, x-m\, _X = \inf_ \, x+m\, _X = \inf_\, y\, _X.

Examples

Let ''C'' ,1denote the Banach space of continuous real-valued functions on the interval ,1with the sup norm. Denote the subspace of all functions ''f'' ∈ ''C'' ,1with ''f''(0) = 0 by ''M''. Then the equivalence class of some function ''g'' is determined by its value at 0, and the quotient space 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

The quotient of a locally convex space by a closed subspace is again locally convex. Indeed, suppose that ''X'' is locally convex so that the topology on ''X'' is generated by a family of seminorms where ''A'' is an index set. Let ''M'' be a closed subspace, and define seminorms ''q''_α on ''X''/''M'' by

:q_\alpha( = \inf_ p_\alpha(v).

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''. p. 54, § 12.11.3

See also

* Quotient group
* Quotient module
* Quotient set
* Quotient space (topology)

References

Sources

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Functional analysis
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Space (linear algebra)