- Quotient space (linear algebra)
In
linear algebra , the quotient of avector 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 anequivalence 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". Theequivalence 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 arewell-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 Wthen 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" isfinite-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 (ornullspace ) of this epimorphism is the subspace "U". This relationship is neatly summarized by theshort 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". Thefirst 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 therank-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 ofseminorm s {"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 aFréchet space , then so is "X"/"M" harv|Dieudonné|1970|loc=12.11.3.ee also
*
quotient set
*quotient group
*quotient module
*quotient space (intopology )References
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