## I.3 Quotients, Functions between Algebraic Structures: Homomorphisms, Isomorphisms, Automorphisms, Linear Maps and Matrices, Eigenvalues and Eigenvectors, Basic Concepts of Mathematical Analysis: Limits; Pages 25 – 30

Define the set Q[x] as the set of all polynomials in the variable x with rational coefficients.  Q[x] is a commutative ring but not a field.  One regards a unique polynomial to be equivalent to the zero polynomial.  Quotients of a set of polynomials allows one to divide the set of all polynomials into equivalence classes where any polynomial “is equivalent to” a polynomial at most some finite degree.  This converts Q[x] into a field.  This creates a way to transform a polynomial with the difference between the original polynomial and the equivalent one a multiple of the unique polynomial, P(x), used to create the equivalence classes.  In the newly created equivalence field all polynomials that are not equivalent to zero (multiples of the unique polynomial) have inverses.  We decide that equivalent polynomials are equal.  The resulting mathematical structure is denote Q[x]/P(x).  It turns out that Q[x]/P(x) is the smallest field that contains Q and also has a root of the polynomial P(x).  It turns out that this is the same field we described earlier as Q(g).

We regarded to objects as equal when the were equivalent.  Formally this is the notion of equivalence classes and equivalence relations.  An example of this is the rational numbers where we regard 1/2 and 2/4 as equivalent to one another.   This means we must be careful when defining binary operations on the equivalence classes.  One must get equivalence objects out when one puts equivalence objects into the binary operation.

A quotient group is defined with G a group and H a subgroup of G.  g1 and g2 elements of G are said to be equivalent if g1-1g2 belongs to H.  The equivalence class of an element g is the set of all elements gh such that h Î H, this is written gH.  It is called a left coset of H.

A natural candidate for a binary operation * on the set of left cosets: g1H * g2H = g1g2H.  One must check that if you pick elements from the cosets g1H and g2H that the product of these elements will indeed be in g1g2H.  This may not be true.  If one has the additional assumption that H is a normal subgroup then it is true.  H is a normal subgroup if when if h is an element of H, then ghg-1 is an element of H for every g of G.  Elements of the form ghg-1 are called conjugates of h.  A normal subgroup is one that is closed under conjugation.

If H is a normal subgroup then the set of left cosets forms a group with the new binary operation abd the groupis written G/Hand is called the quotient subgroup of G by H.

It turns out that a torus can be described as a quotient subgroup of R2 by H.  For a H that is an integer translation of points in R2.    Other examples of this are fundamental groups, homology and cohomology groups and a moduli space.

There are interesting functions between algebraic structures.  Some basic structures are homomorphisms, isomorphisms and automorphisms.  Particular mathematical structures include: groups, fields and vector spaces.  Consider the class of functions between two mathematical structures X and Y.  Functions that preserve structure are interesting.  If there is structure between elements of X, say a and b, and this structure is preserved between the images of these elements, say f(a) and f(b) (elements of Y) then the structure is preserved.

Examples of this structure might be ab = c and the preserved structure is f(a)f(b) = f(c) with X and Y groups.  Or, this same strucure if X and Y are fields.  Also, if X and Y are fields we may have the additive structure f(a) + f(b) = f(c) whenever a + b = c.  Vector space structure to be preserved is linear combinations, so that, f(av + bw) = af(v) + bf(w).

A function that preserves structure is called a homomorphism.  A homomorphism of vector spaces is called a linear map.  A homomorphism can collapse structure.

An isomporphism is a homomorphism that has an inverse that is also a homomorphism.  For most algebraic structures if f has an inverse g then g is automatically a homomorphism.  In such cases an isommorphism that is also a homomorphism that is also a bijection.If there is an isomorphism between two algebraic structures X and Y then X and Y are said to be isomorphic.  Isomorphic menas “the same in all essential respects”.  The kind of objects in the sets X and Y are not essential.

An automorphism is an isonmorphism from X to itself.  What is important about automorphisms is the algebraic symmetries induced in the group.  The composition of two automophisms is another automporphism.   The automorphisms of a structure X form a group.

An example of automorphisms is given with Q(2).  There are two automorphisms  that form a group consisting of the elements +1, -1 under multiplication.  This is also the group of integers modulo 2, the group of symmetries of a non equilateral isosceles triangle and many more.

Kernels of a homomorphism between tow algebraic structures are the set of elements of X such that the image of the element is the identity of Y.  (Additive if both addition and multiplication are the binary operations.)  If G and K are groups then kernel of a homomorphism form G to K is a normal subgroup of G.  If H is a normal subgroup of G then the quotient map which takes each element of G to the left coset gH is a homomorphism from G to the quotient group G/H with kernel H.  The kernel of any ring homomorphism is an ideal.  every ideal I in a ring R is the kernel of a quotient map from R to R/I.

Linear maps preserve linear combinations in vector spaces.  These can be written as matrices with the usual definition of matrix multiplication.  The linear maps that can be inverted are automporphisms.  There are also infinite dimension linear maps, such as, differentiation and integration.

One can determine eigenvalues and eigenvectors for linear maps.

One of the basic concepts of mathematical analysis is that one can specify an object indirectly as the limit of objects that one can list directly.  One uses better and better approximations.  This is the basis of calculus.

Limits of sequences can approach a limit.