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").

Definition

Formally, the construction is as follows harv|Halmos|1974|loc=§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" − "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" with "N" being the zero class, [0] .

Examples and properties

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". This gives one way in which to visualize quotient spaces geometrically.

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 ("x"₁,…,"x"_"n"). The subspace, identified with R^"m", consists of all "n"-tuples such that only the first "m" entries are non-zero: ("x"₁,…,"x"_"m",0,0,…,0). Two vectors of R^"n" are in the same congruence 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.

More generally, if "V" is written as an (internal) direct sum of subspaces "U" and "W"::$V=Uoplus\; W$then the quotient space "V"/"U" is naturally isomorphic to "W" harv|Halmos|1974|loc=Theorem 22.1.

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" harv|Halmos|1974|loc=Theorem 22.2::$mathrm\{codim\}(U)\; =\; dim(V/U)\; =\; dim(V)\; -\; dim(U).$

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:$0\; o\; U\; o\; V\; o\; V/U\; o\; 0.,$

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:$|\; [x]\; |\_\{X/M\}\; =\; inf\_\{m\; in\; M\}\; |x-m|\_X.$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".

Generalization to locally convex spaces

The quotient of a locally convex space by a closed subspace is again locally convex harv|Dieudonné|1970|loc=12.14.8. Indeed, suppose that "X" is locally convex so that the topology on "X" is generated by a family of seminorms {"p"_α|α∈"A"} where "A" is an index set. Let "M" be a closed subspace, and define seminorms "q"_&alpha by on "X"/"M"

:$q\_alpha(\; [x]\; )\; =\; inf\_\{xin\; [x]\; \}\; p\_alpha(x).$

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" harv|Dieudonné|1970|loc=12.11.3.

ee also

*quotient set
*quotient group
*quotient module
*quotient space (in topology)

References

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