Understanding Quotient Spaces in Mathematics
In group theory, a subgroup H of a group G helps decompose G into equal-size disjoint subsets called cosets. Quotient spaces in mathematics involve equivalence classes under a given relation and a specific topology. Furthermore, in linear algebra, the quotient of a vector space by a subspace results in a new vector space. Explore the concept of quotient spaces and their applications in various mathematical contexts.
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QUOTIENT SPACES CHAPTER 3
In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by [G : H]. COSETS
The quotient space of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation ) together with the following topology given to subsets of : a subset of is called open iff is open in . Quotient spaces are also called factor spaces. Quotient space
Suppose if A is group, and B is subgroup of A, and is an element of A, then aB = {ab : b an element of B } is left coset of B in A, The left coset of B in A is subset of A of form aB for some a(element of A). In aB(left coset), a is representative of coset. And Ba = {ba : b an element of B } is right coset of B in A. The right coset of B in A is subset of A of form Ba for some a(element of A). In right coset Ba, element a is referred to as representative of coset.
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"). DIMENSION S OF A QUOTIENT SPACE
Formally, the construction is as follows.[1]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. 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 : 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[2][3] [x] = [ x] for all 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 representatives). 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. Alternatively phrased, the quotient space V/N is the set of all affine subsets of V which are parallel to N.
