dansmath > lessons > 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

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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)

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Vector Spaces (top of page)

The "canonical example" of a vector space is the Euclidean plane R2, 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 = (v1, v2, . . . , vn) and w = (w1, 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) where 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..

 

(back to Lin Alg)

 

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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?

 

 

(back to Lin Alg)

 



 

Differential Equations (expanded 8/01) (top of page)

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 = dv/dt = d2s/dt2 ; there's the derivative.

 



 

Abstract Algebra - Groups, rings, fields (8/01) (top of page)

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

 

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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!]

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

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.

 

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Rings and Ideals (top of page)

 

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) with

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 2Z of even integers is an ideal of Z.

 

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Fields and finite extensions of the rationals (top of page)

 

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.

 

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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 correspond'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!

 

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