Quotient Space (linear Algebra)
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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 (Halmos 1974, §21-22). 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 − yN. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce 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 often denoted

[x] = x + N

since it is given by

[x] = {x + n : nN}.

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] = [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 with N being the zero class, [0].

The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map.

Examples

Let X = R2 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 one way in which 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.)

Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1,...,xn). The subspace, identified with Rm, consists of all n-tuples such that the last n-m entries are zero: (x1,...,xm,0,0,...,0). Two vectors of Rn are in the same congruence class modulo the subspace if and only if they are identical in the last nm coordinates. The quotient space Rn/ Rm is isomorphic to Rnm in an obvious manner.

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

then the quotient space V/U is naturally isomorphic to W (Halmos 1974, Theorem 22.1).

An important example of a functional quotient space is a Lp 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

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 (Halmos 1974, Theorem 22.2):

Let T : VW be a linear operator. The kernel of T, denoted ker(T), is the set of all xV 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 : VW 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

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] with the sup norm. Denote the subspace of all functions fC[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.

Generalization to locally convex spaces

The quotient of a locally convex space by a closed subspace is again locally convex (Dieudonné 1970, 12.14.8). Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα | α ∈ A} where A is an index set. Let M be a closed subspace, and define seminorms qα by on X/M

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 (Dieudonné 1970, 12.11.3).

See also

References

  • Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 978-0-387-90093-3.
  • Dieudonné, Jean (1970), Treatise on analysis, Volume II, Academic Press.

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