Goto: Prof. King's page at Univ. of Florida.
Or: JK Homepage.

Modified:
Tuesday, 02Aug2016
Printed:
Sunday, 17Nov2019

Page:
http://squash.1gainesville.com/Include/thispage.shtml

Aut2008: MAS3114 3241 Computational.Linear.Alg MWF7 LIT203

(Note: Our classroom is on the 2^{nd} floor of Little Hall, East wing, North side.)

*:: High-school knowledge*
(formula for a line between two points, the Quadratic Formula,
intersecting a line with a parabola, etc.), and some

*:: Sets&Logic ideas*
(Equivalence relations, partial orders, binary operators,
induction, pigeon-hole principle, cardinality).

*:: Mathematical maturity*
(Do you remember your basic calculus stuff? Do you remember how
to sum a geometric series?)

:: Math-Greek alphabet (pdf),
which we will use in class frequently.

This is a 1-semester course for folks interested in Mathematical Cryptography
(major topic) and other aspects of *Coding*:
Data Compression and (*time permitting*)
Error-correcting codes.
We may also cover one example of an Isomorphism code.

It is accessible to anyone who has Sets&Logic knowledge, whether or not you've take that course. I would like you to know:

*What a prime number is*,
and *What mathematical induction is*
and *What an equivalence relation is*.
Also helpful is how to “add two numbers mod N”,
and what the Euler phi function is.

Good choices [all these books are in Marston Science Library, on campus] for a bit of self-study are:

- Elementary Number Theory by James Strayer;
*or* - A Friendly Introduction Number Theory by Joseph Silverman;
*or* - Elementary Number Theory by David Burton;
*or* - the text by Vanden Eynden.

Our Teaching Page
has useful information for students in all of my classes.
It has **my schedule**,
LOR guidelines,
and Usually Useful Pamphlets.
One of them is the
*Checklist* (pdf)
which gives pointers on what I consider to be good mathematical
writing.
Further information is at our
class-archive URL
(I email this private URL directly to students).

The following came from
Earliest Known Uses of Some of the Words of Mathematics.

ALGEBRAcomes from the title of a work written in Arabic about 825 by al-Khowarizmi, al-jabr w'al-muqabalah, in which al-jabr means "the reunion of broken parts." When this was translated from Arabic into Latin four centuries later, the title emerged as Ludus algebrae et almucgrabalaeque.In 1140 Robert of Chester translated the Arabic title into Latin as Liber algebrae et almucabala.

In the 16th century it is found in English as

algiebar and almachabel, and in various other forms but was finally shortened toalgebra. The words meanrestoration and opposition.In Kholâsat al-Hisâb (Essence of Arithmetic), Behâ Eddîn (c. 1600) writes,

The member which is affected by a minus sign will be increased and the same added to the other member, this beingalgebra; the homogeneous and equal terms will then be canceled, this beingal-muqâbala.The Moors took the word

al-jabrinto Spain, analgebristabeing a restorer, one who resets broken bones. Thus in Don Quixote (II, chap. 15), mention is made ofAt one time it was not unusual to see over the entrance to a barber shop the wordsun algebristawho attended to the luckless Samson.Algebrista y Sangrador[bonesetter and bloodletter] (Smith vol. 2, pages 389-90).The earliest known use of the word

algebrain English in its mathematical sense is by Robert Recorde in The Pathwaie to Knowledge in 1551:Also the rule of false position, with dyvers examples not onely vulgar, but some appertayning to the rule of Algeber.The phrase

an algebrais found in 1849 Trigonometry and Double Algebra by Augustus de Morgan:Ordinary langauge has methods of instantaneously assigning meaning to contadictory phrases: and thus it has stronger analogies with an algebra (if there were such a thing) in which there are preorganized rules for explaining new contradictory symbols as they arise, than with one in which a single instance of them demands an immediate revision of the whole dictionary[University of Michigan Historical Math Collection]. …

Real Analysis is no more about reality than Complex Analysis is about complexity.-P. Boyland, paraphrased

The following is from
Wikipedia, the free encyclopedia

Cryptography(orcryptology; derived from Greekκρυπτός kryptós"hidden," and the verbγράφω gráfo"write") is the study of message secrecy. In modern times, it has become a branch of information theory, as the mathematical study of information......

Steganography(i.e., hiding even the existence of a message so as to keep it confidential) was also first developed in ancient times. An early example, from Herodotus, concealed a message - a tattoo on a slave's shaved head - by regrown hair. More modern examples of steganography include the use of invisible ink, microdots, and digital watermarks to conceal information .Ciphertexts produced by classical ciphers always reveal statistical information about the plaintext, which can often be used to break them. After the Arab discovery of frequency analysis (around the year 1000), nearly all such ciphers became more or less readily breakable by an informed attacker. ... Essentially all ciphers remained vulnerable to cryptanalysis using this technique until the invention of the polyalphabetic cipher by Leon Battista Alberti around the year 1467. Alberti's innovation was to use different ciphers (ie, substitution alphabets) for various parts of a message (often each successive plaintext letter). He also invented what was probably the first automatic cipher device, a wheel which implemented a partial realization of his invention. In the polyalphabetic Vigenère cipher, encryption uses a key word, which controls letter substitution depending on which letter of the key word is used.

The following is abridged from
Wikipedia, the free encyclopedia

Diophantus of Alexandria- (Greek:Διόφαντος ὁ Ἀλεξανδρεύς, circa200/214– circa284/298) was a Greek mathematician of the Hellenistic era. Little is known of his life except that he lived in Alexandria, Egypt ...He was known for his study of equations with variables which take on rational values and these Diophantine equations are named after him. Diophantus is sometimes known as the

father of Algebra. He wrote a total of thirteen books on these equations. Diophantus also wrote a treatise on polygonal numbers.In 1637, while reviewing his translated copy of Diophantus' Arithmetica (pub. ca.250) Pierre de Fermat wrote his famous

Last Theoremin the page's margins. His copy with his margin-notes survives to this day.Although little is known about his life, some biographical information can be computed from his epitaph (see links below). He lived in Alexandria and he died when he was 84 years old. Diophantus was probably a Hellenized Babylonian.

A 5

^{th}and 6^{th}century math puzzle involving Diophantus' age: He was a boy for one-sixth of his life. After one-twelfth more, he acquired a beard. After another one-seventh, he married. In the fifth year after his marriage his son was born. The son lived half as many as his father. Diophantus died 4 years after his son. How old was Diophantus when he died?

The answer: 84 The answer is determined from two methods: 1. Finding the common multiple of 12, 6, and 7 (which is 84). 2. Taking 14 (the age up to which would be considered a boy; one-sixth of his life) multiplied by 6, which equals 84.

- "How To Write Proofs" (html), by Prof. Larry W. Cusick, [INTERMEDIATE]. Examples mostly from Elem. Number Theory; some from Calculus.
- Prof. Christopher Heil's page (pdf) [4 pages, INTRO]. A well written survey of the structure of proofs. Has one example of induction (recursion).
- “How to write proofs: a quick guide” by Prof. Eugenia Cheng (pdf)
[17 pages, INTRO].
Good, friendly. She is stricter than I in the ordering of steps in a proof.
Conversely, her write-up is lax in places where I am strict:
- Always use a
*word/phrase*, and never a comma, forthen

; e.gthen

ornecessarily

orwe discover to our delight that

. - She has proof snippets in her examples, but remember
that
*your*proofs must be written as a sequence of complete, grammatical sentences, correctly punctuated, structured into coherent paragraphs.

- Always use a

This branch of mathematics is the only one, I believe, in which good writers frequently get results entirely erroneous. ... It may be doubted if there is a single extensive treatise on probabilities in existence which does not contain solutions absolutely indefensible.C. S. PEIRCE,

inPopular Science Monthly(1878).

I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives.-Charles Hermite

There's a delta for every epsilon, It's a fact that you can always count upon. There's a delta for every epsilon And now and again, There's also an N. But one condition I must give: The epsilon must be positive A lonely life all the others live, In no theorem A delta for them. How sad, how cruel, how tragic, How pitiful, and other adjec- Tives that I might mention. The matter merits our attention. If an epsilon is a hero, Just because it is greater than zero, It must be mighty discouragin' To lie to the left of the origin. This rank discrimination is not for us, We must fight for an enlightened calculus, Where epsilons all, both minus and plus, Have deltas To call their own.Words and Music by Tom Lehrer,

American Mathematical Monthly, 81 (1974)

The following is from
Wikipedia, the free encyclopedia

Linear Algebra– Linear algebra stems from the need to solve systems of linear equations. For small systems,ad hocmethods are sufficient. Larger systems require one to have more systematic methods. The modern day approach can be seen 2,000 years ago in a Chinese text, the Nine Chapters on the Mathematical Art (Traditional Chinese: 九章算術; Simplified Chinese: 九章算术; pinyin: Jiǔzhāng Suànshù).Chinese mathematicians developed a system in which they organized linear equations in a rectangular pattern called Fāng Chéng (方程) in Chinese, involving horizontal and vertical counting rods. This rectangular representation of linear equations is the equivalent of today's matrix.

One of the key developments in linear algebra was the modern day method of solving linear systems known as Gauss-Jordan elimination, after German mathematician Carl Friedrich Gauss (1777-1855) and German engineer Wilhelm Jordan (1844-1899). Gauss called the method

eliminatio, even though the Chinese were using an almost identical method nearly two millennia prior. This method stemmed really from Gauss's laziness in leaving off variable stems such asx1, x2, etc. in solving largen-tuples of linear equations while following the asteroid now known as Ceres. His method is explained in his book Theoria Motus Corporum Coelestium (1809).

The following is from
Earliest Known Uses of Some of the Words of Mathematics

NUMBER THEORY.According to Diogenes Laertius, Xenocrates of Chalcedon (396 BC - 314 BC) wrote a book titled The theory of numbers.A letter written by Blaise Pascal to Fermat dated July 29, 1654, includes the sentence,

The Chevalier de Mèré said to me that he found a falsehood in the theory of numbers for the following reason.The term appears in 1798 in the title Essai sur la théorie des nombres by Adrien-Marie Legendre (1752-1833).

In English,

theory of numbersappears in 1811 in the title An elementary investigation of the theory of numbers by Peter Barlow.

Number theoryappears in 1853 in Manual of Greek literature from the earliest authentic periods to the close of the Byzantine era by Charles Anthon: "The ethics of the Pythagoreans consisted more in ascetic practice and in maxims for the restraint of the passions, especially of anger, and the cultivation of the power of endurance, than in scientific theory. What of the latter they had was, as might be expected, intimately connected with their number-theory" [University of Michigan Digital Library].

Number theoryappears in 1864 in A history of philosophy in epitome by Dr. Albert Schwegler, translated from the original German by Julius H. Seelye: "Not only the old Pythagoreans, who have spoken of him, delighted in the mysterious and esoteric, but even his new-Platonistic biographers, Porphyry and Jamblichus, have treated his life as a historico-philosophical romance. We have the same uncertainty in reference to his doctrines, i. e. in reference to his share in the number-theory. Aristotle, e. g. does not ascribe this to Pythagoras himself, but only to the Pythagoreans generally, i. e. to their school" [University of Michigan Digital Library]. ...

**At all times** have a **paper copy** you can hand-in; I do
** NOT** accept
electronic versions.
Print out a copy

My dog ate my homework.)

Please follow the guidelines on the
*Checklist*
(pdf, 3pages) to earn full credit.

Quantifiers ∀ and ∃ (“for all”

and“there exists”) are like nitroglycerin, in that one little mis-step leads to the whole thing blowing up in your face.There is no partial credit when it comes to Explosives and Quantifiers.

-JLF King

We are sorry, but the number you have dialed is imaginary. Please rotate your phone by 90 degrees and try again.-David Grabiner

The four-year-old niece of a mathematician was playing a game in which she was the conductor on a train and her mother was a passenger.Wait a minute,said Nancy,we have to get some paper to make tickets.Oh,said her mother, who had probably had a long day,do we really need them? After all, it's only a pretend game with pretend tickets.

Oh no, Mommy, you're wrong,replied Nancy;they're pretend tickets, but it's arealgame.

(recounted by David Gale. This appeared

inThe Intelligencer.)

I will be running a Special Topics NT course for Fall 2007, as long as it has at least 14 students.

This course does NOT require NT 1 (MAS4203).
[For those who have already taken MAS4203, I have designed this course
to be a good successor.]
For students who have ** not** already taken MAS4203,
I ask that (during the summer) they
read about 30 pages from
Shoup's free online text
[linked below].

This course will counted as a math-major elective and as one of the three 4000-level math courses required for the math minor, but ISIS will not know this; you'll have to ask the Undergraduate Coordinator to adjust your audit after the fall semester has begun.

If potentially interested in taking the course, please email me at

squashATuflDOTedu (Prof. JLF King)

and I will add you to the mailing list. Most of the information about the
course will be distributed by email.

Our course is a 3 credit course.

I was just informed that one must register at the Math department for any of the Special Topics courses.

I was just informed that one must register at the Math department for any of the Special Topics courses.

As you can see on my
Schedule
I will have my 2006 office-hours on 7^{th}-period. I also have
OH by appointment, but if you are able to keep one of M,W,F-7^{th}
free for office-hours, that would be convenient for you and me.

Welcome!
Our Teaching Page
has useful information for students in all of my classes.
It has **my schedule**,
LOR guidelines,
and Usually Useful Pamphlets.
One of them is the
*Checklist* (pdf)
which gives pointers on what I consider to be good mathematical
writing.
Further information is at our
class-archive URL
(I email this private URL directly to students).

In all of my courses, attendance is absolutely required (excepting illness and religious holidays). In the unfortunate event that you miss a class,you are responsibleto get all Notes / Announcements / TheWholeNineYards from a classmate, or several. All my classes have a substantial class-participation grade.

A syllabus (text file) is available. On the first day of class I will hand-out a paper syllabus, which has the instructions on how to use the class archive. The archive is at a private URL, only for the use of the students in our class.

Whatever you do, *Don't look at*
Past courses with notes,
exams and links!

There
~~ will be ~~
** was** a

Friday., 13Jan., is
the *Last day* of **Add/Drop**,
so that students can self-evaluate if they should be in the course.
The mini-test will count for 6%—8% of course grade.

The mini-test will select *some* topics from:

*High-school knowledge*(formula for a line between two points, the Quadratic Formula, intersecting a line with a parabola, etc.)- The Math-Greek alphabet (pdf), which we will use in class frequently.
- Calc 1 knowledge (Integration by parts, Intermediate-Value Thm and Mean-VT. Chain rule, etc.)

If you missed the prereq exam, you must make it up
posthaste; Thursday midafternoon at the latest.

Here is a
sample test (pdf)
which is longer than the actual one will be.
Note that this is an
**open brain
closed text,
no calculator
**
exam.

There was a test of prerequisite knowledge (pdf) on Friday, 07Jan2011. Our mini-test will count for 7%—9% of course grade.

If you missed the prereq exam, you must make it up posthaste. Please email me immediately at squashATuflDOTedu (Prof. Jonathan King)
Tuesday, 11Jan2011,
is the *Last day*
of **Add/Drop**; students can self-evaluate if they
should be here, or if they should transfer to
Intro Number Theory,
MAS4203 8430,
*MWF4 LIT219, Prof. Alladi*.

The mini-test will examine beginning NT topics,
*Modular arithmetic, Euler-phi fnc, Chinese Remainder Theorem*
as well as *basic facts on primes*
and a beginning-level *NT proof* or *Induction proof*.

It will also select topics from your Calc 1,2 knowledge (Differentiation rules, Integration, basic series), and High-school knowledge (formula for a line between two points, the Quadratic Formula, intersecting a line with a parabola, etc.) and the Math-Greek alphabet (pdf), which we will use in class frequently.

Here is a
sample test (pdf)
which is longer than the actual one will be.
This is an
**
open brain,
closed text,
no calculator
**
exam.

There will be a **test of prerequisite knowledge**
Monday, 27Aug2007.
Observe that Wedn., 29Aug, is our *Last class-day*
of **Add/Drop**, so that students can self-evaluate if they
should be in the course.
The mini-test will count for 6%—9% of course grade.

The mini-test will select *some* topics from your Calc 1-3 knowledge
(Integration by parts,
Intermediate-Value Thm and Mean-VT.
Multi-variate chain rule,
Taylor series,
Multiple integrals,
etc.),
and High-school knowledge
(formula for a line between two points, the Quadratic Formula,
intersecting a line with a parabola, etc.)
and the
Math-Greek alphabet (pdf),
which we will use in class frequently.

- Homework-2 (txt), due Monday, 05Nov2007.
- The thought-provoking
Class-B (pdf)
has been instantiated.
Astonishingly, one earns genuine coin-of-the-realm CP points (CP = "class participation") by posting a solution or improving a solution.

- Applause and acclaim greeted the gorgeous Prereq-A (pdf) with its clean lines, and minimalistic two-tone format; this was the test of prerequisite knowledge.

The various Number Theory czars who help out.

Projector | Phone-list | Chalk | Blackboard | H-Probs | Time |
---|---|---|---|---|---|

? | ? | Charlye | Marshall | Jimmy-C & Dream | Cameron |

Our textbook is
Math
(**Third** Edition **Update**).

Here are links to
this book at The Publisher's site
and at
Amazon.com.

Author: | David Lay | Year: | 2006 |

ISBN: | 0-321-28713-4 | Publisher: | Addison-Wesley |

There will be ** not be a final exam** .
Rather, there will be an

*The final project must be carefully typed.*
I recommend learning the free
mathematics-typesetting language
.
(It is the archive language of the American Mathematical
Society.)
It can be learned in a week. It is very good for typesetting homework.

**At all times** have a **paper copy** you can hand-in; I do
** NOT** accept
electronic versions.
Print out a copy

My dog ate my homework.)

Please follow the guidelines on the
*Checklist*
(pdf, 3pages) to earn full credit.

Teaching Page: http://squash.1gainesville.com/teaching.html

Footnotes: http://squash.1gainesville.com/info.footnote.html