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**beyond calculus** - Linear Algebra (Matrices, systems of linear eqns, vector spaces)
- Differential Equations (Definitions, solution methods, applications)
- Abstract Algebra (Groups, Rings, Fields, Galois Theory)
- Higher Math (for math majors!) (Real & complex analysis, topology, ...)
- (c) 1997-2001 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.
**Linear Algebra**11/97 expanded 8/01- This is a very interesting subject that bridges the gap from basic algebra (systems of linear equations) to geometry (dot and cross products) to abstract algebra (vector space axioms).
- Matrices . . . . . Vector Spaces . . . . . Other Lin Alg Things

**Matrices**(top of page)- The underlying theme is
**m**x**n****matrices**, which are rectangular arrays of numbers with m rows and n columns. We can**add**matrices if they're the same size (ex. 1), multiply all entries by a constant (ex. 2), and even multiply matrices if they're "compatible" (ex. 3). **Example 1.**A pair of 3 x 2 matrices can be**added**, A + B = C (position by position):**3****8****7****9****10****17****Matrix****Addition****- 4****0****+****5****2****=****1****2****7****- 10****1****12****8****2****Example 2.**We can**multiply**all entries of B**by a scalar**k :to get k B**7****9****28****36****Scalar****Multip****4*********5****2****=****20****8****1****12****4****48****Example 3.**If A is m x**n**and D is**n**x p then we can**multiply**A * D (answer is m x p)**3****8****3**9+32 **41****Matrix****Multip****- 4****0*********4****=**-12+0 = **- 12****7****- 10**21-40 **- 19**- Try to follow the multiplications here; go across on A and down on D.
- The rows of A are 'dotted' with the columns from D, if you know what I mean.
- (back to Lin Alg)

**Vector Spaces**(top of page)- The "canonical example"
of a
**vector space**is the**Euclidean plane R****2**, the pairs - (x, y) of real numbers, with
componentwise
**addition**and**scalar multioplication****.** - But any set V with those properties is a vector space:
- A common operation in Rn
(Euclidean n-space) is the
**dot product:** - if
**v = (v****1, v2, . . . , vn)**and**w = (w****1, w2, . . . , wn) ,**then we define **v . w = v1 w1 + v2 w2 + . . . + vn wn . Beware:**This isn't the only dot-like- product in Rn, and not every vector space has a dot product.
- A nice property of this is
that if the
**length**of**v**is defined as**| v | = (v . v) ,** - we have the useful formula
**v . w = | v | | w | cos(t)****angle**between**v**and**w**. **Examples of vector spaces**- a) The set of all triples (x, y, z) of real nos ; the usual Cartesioan 3-space.
- b) The set of all 3 x 2 matrices formes a vector space under + and scalar *.
- c) The set of polynomials p(x) of degree n or less ; using p + q and k p.
- d) The set of continuous functions f from [0, 1] --> R. Add & sc. mult..

**Other Linear Algebra Things**(top of page)- Further areas of study:
**linear transformations**from one vector space to another, **matrix, rank**and**nullity**of a lin. transf,**determinants,**calculating area & volume,**inner products**such as the dot product, and**eigenvalues and eigenvectors**and**factor decomposition**and**diagonalization**of matrices. Got all that?**Differential Equations**(expanded 8/01)- How does a population grow? What about interest compounded in a bank account?
- These quantities grow at a rate proportional to their current size (as in "4% per year").
- If we let
**y(t)**= the**amount**(of people or dollars)**at time t**(years) , then the growth is - (*)
**y'(t) = k y(t)**or just**y' = k y**, where k = the "growth constant." - This relationship between
a function and its derivative is called a
**differential equation**. - In this case we write
**y'(t) = dy / dt**and rewrite (*) as :**dy = k y dt**; so that **dy / y = k dt**; integrate and get :**ln |y| = k t + c**; so that**|y| = e^(kt + c) = e^c e^(kt)**; so**y = (+ or - e^c) e^(kt)**; thus**y = A e^(kt)**; where A = (pos or neg) initial amount.- The function
**y(t) = A e^(kt)**is called the**general solution**of the diff eqn (*). - With appropriate
**initial conditions**we can get a**particular solution**to a diff eqn. **Example :**If we know that a population of**P(t)**people at time t grows at a rate that's- proportional to its current
size, then
**P'(t) = k P(t) ;**if we know the growth rate k - where k = (birth rate + immigration rate) - (death rate + emigration rate), and we
- know another fact, say the
initial population P(0), we have the
**exponential model** **P(t) = P(0) e^(k t) ;**a population starting at 5000 and growing at 3% per year,- say, would have
**5000 e^(.03 t)**people after t years. P(10) = 6749 for instance. **Other areas of study**in diff eqs include**vector fields**and solution curves, soluition of- motion problems in
**physics**by analyzing**forces**, and applying Newton's Law**F = m a** - where
**F**= force,**m**= mass,**a**= acceleration = d**v**/dt = d2**s**/dt2 ; there's the derivative. **Abstract Algebra**-**Groups, rings, fields**(8/01)- Dan's personal co-favorite subject (see also Number Theory).
- Groups, Abelian and Non-Abelian
- Rings and Ideals, with examples
- Fields, Extensions of the Rationals
- Galois Theory, Galois Groups

**Groups**(top of page)- A
**group**is a set G adorned with an operation * (usually omitted like multiplication) - such that G is 'closed' under *, that * is associative and G has an identity and inverses.
**Order:**If G has a finite number n of elements we say 'n is the order of G' or 'G has order n'.**Closed**: This means the product g*h = gh of any two elements g and h of G is still in G.**Associative**: The equality holds: g*(h*k) = (g*h)*k , for all g,h,k in G [short: g(hk) = (gh)k]**Identity**element: There is a (unique) element e in G with e*g = g*e = g for all g in G.- The uniqueness is math's shortest proof: If e, f are identities then e = ef = f.
**Inverse**elements: For each g in G there is a unique elt h in G with g*h = h*g = e.- The inverse h of g is called 'g inverse' and is denoted g^(-1). [This is NOT 1/g.]

Examples of groups:

- The set of reals, rationals,
or integers under addition; {
**R**, +},{**Q**, +},{**Z**, +}.- In all three cases, the inverse
of
**n**is -**n**[Check all the properties!]

- In all three cases, the inverse
of
- The positive rationals under
multiplication: {
**Q**+, *} (Inverse of x**is**1/x in this case!) - The set of vectors in the
plane (or 3-space) (same
as {
**R**, +} position-wise, 2 or 3 times.) - The set of m x n matrices
under addition (same as
{
**R**, +} position-wise, mn times.) - The set of functions f from
**R**to**R**, say, under function composition. - A group might be commutative, called 'Abelian', or it might be 'non-Abelian'..
**Abelian / Commutative:**For all g and h in G, we have gh = hg.-
The groups {

**R**, +},{**Q**, +},{**Z**, +} are all Abelian because a + b = b + a. - The identity element is
**I**= {{1, 0},{0, 1}}; inverses work out nicely, it's associative, but - the two matrix products AB and BA are very often different (you look for an example)
- There are many other types
of groups:
**finite**groups (not infinite as sets),**p-groups**(groups of size p^k), **normal subgroups, symmetry groups**of geometric objects,**permutation groups**,**Galois groups**.

**Rings**and Ideals- A
**ring**is a set R (not the real numbers**R**) with**two**operations, + and * , such that: - R is closed under both ops, R is an abelian group under +, and has distributive laws.
- Most rings dealt with at
first are
**commutative rings with identity**(for multiplic'n). - Some famous rings: the 'ring
of integers'
**Z,**(or rationals**Q**, reals**R**, complex nos**C**) - usual addition and multip. For example, the set M of 3 by 3 matrices forms a ring under
- addition A+B and matrix multiplication AB. Scalar multiplication kA isn't used in this case.
- A subset S of a ring R that's
also a ring is called a
**subring**; I write S < R for subring. - A subset J of R is an
**ideal**of R if J is an additive subgroup of R, and if all products**j****r** - (with j in J, r in R), are
**in J**. For example the subring 2**Z**of even integers is an ideal of**Z**.

**Fields**and finite extensions of the rationals- Just as non-zero numbers have reciprocals and square matrices (with determ. =/= 0)
- have inverses, we define
a
**field**as a set K that's a ring as above but with the extra - condition that all nonzero elements x have an inverse x^(-1) ; general notation, not
- nec. reciprocals or inverse functions, although those are some examples!
- If K is any field, a
**field extension of K**is a field L that contains K. - For example,
**R**is an extension of**Q**because all rationals are real. - Also the field K =
**Q(i)**= {a + b i , i^2 = -1, a, b in**Q**} is an extension of**Q**. - The second case is an example
of a
**finite extension of Q**meaning that all numbers - in K can be written as linear combinations of 2 things: 1 and i.
- This is the same as the dimension
of K as a vector space over
**Q**(see Linear Algebra). - The
**minimal polynomial**of an element b in K over**Q**is the smallest degree p(x) - such that p(b) = 0. For example
the min poly for i over
**Q**is p(x) = x^2 + 1; p(i) = 0.

**Galois Theory**(top of page)- This is a beautiful connection between finite groups of order n and finite extensions of degree n of fields.
- In fact the key is the minimal polynomial f(x) of an element of L over K (see above) and how it factors.
- For many (finite) extensions L over K, there is a (finite) group G of all K-automorphisms of L meaning
- 1-1, onto functions f: L --> L with: f(a+b) = f(a) + f(b), f(ab) = f(a) f(b), all a, b in L, f(k) = k, all k in K.
- This group G is called the
**Galois group**of the extension L over K. Not all extensions are 'Galois'. - The notion of 'fixed field': the set of autos of L that fix (leave f(a) = a) fields M larger than K form a
- subgrp H of G;
**subfields**M < L**and****subgroups**H < G**form a 1-1 corresp**ond'ce under 'fixedfieldof'. - Galois also proved that some polynomials of degree 5 or more were unsolvable by radicals, meaning
- their roots are not expressible as a finite series of the 4 basic operations along with exponents & roots.
- This boils down to showing the Galois group of the splitting field of the polynomial is a 'solvable group'.
- The results of Galois Theory also settled the three classical Greek geometric construction problems: that
- 1) squaring a circle, 2) doubling a cube, 3) trisecting an angle, are all impossible with ruler & compass.
**Higher Math**(math major stuff)**Real Analysis -**Starts where delta & epsilon leaves off. Rigorous study of limits, convergence of sequences and series, new "metrics" (ways of measuring distance), all kinds of cool stuff like compact spaces and uniform continuity**Complex Analysis -**Rigorous study of functions, limits, convergence of series of complex numbers, analytic functions, conformal maps, contour integrals, fundamental theorem of algebra (every poly has a root in**C**), applic's to electrical engineering**Topology -**The study of how coffee cups are just like donuts; each is a solid blob with one hole in it. Distances and angles are ignored in favor of "connectedness" and "smoothness." Spheres can be turned inside-out without tearing holes!**Differential Geometry -**More like calculus on surfaces; this is an extension of Vector Calculus (DVC's Math 292). Ever notice that an inner tube ("torus") looks locally just like the xy-plane? (Think of a patch on the inner tube.)**Well, that's it for now. Check back often for new stuff!****Click below for other topics, or visit the ask dan page!****[ diff eqns | top of page | lessons index]**+ Basic Skills + Arithmetic, Prealgebra, Beginning Algebra + Precalculus + Intermed.Algebra, Functions & Graphs, Trigonometry + Calculus + Limits, Differential Calc, Integral Calc, Vector Calc + Beyond Calculus + **You Are Here**Linear Algebra, Diff Equations, Math Major stuff + Other Stuff + Statistics, Geometry, Number Theory **[ home | info | meet dan | ask dan | dvc ]**

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The group M of 2x2 matrices (with non-zero determinant) under matrix mult. is NOT Abelian.