Solutions to Insomnia

To my mind this title is really punny...at least to the first part of this note.

It is 2:13 am. I've decided to pull and all-nighter, except technically it isn't an all-nighter because I think I took a 3 hour nap earlier today. I took a 2 hour nap and a 1 hour nap the two preceeding days and I really don't like the trend that this is going in. Clearly, I am tired - I'm having trouble stringing a coherent and gramatically correct sentence together, and it has taken me nearly 10 minutes to type this. But if I go back to sleep...

1) I'll fall asleep, then sleep and sleep until noon and then I'll lose much of the day and I'll be annoyed and stay up late saturday night and sunday night as well.

2) I'll be tired but I won't be able to fall asleep because there's too much light [there is no light.] or I just won't be able to sleep.

So I'm going to stay awake; however, I need something to do. Most people are asleep. And quite frankly I doubt my ability to hold a conversation with anyone that I know. Recently Borders had this really awesome book clearance, on account of their filing for bankruptcy [which is not so awesome] and I found this nice-looking book on linear algebra. Except nice-looking books on linear algebra aren't what I usually read.

And this is why I think the title is punny.

--

2:38 am

Linear Algebra by Georgi E. Shilov

A Summary [I'll either learn something or wake up with my face plastered to the keyboard with the wisps of a really bizarre dream about numbers.]:

Preface:

"it should be noted

that the term 'linear algebra' has for some time ceased to describe the actual

content of the course, representing as it does a synthesis of various ideas from

algebra, geometry and analysis. And although analysis in the strict sense of the

term (i.e., the branch of mathematics concerned with limits, differentiation,

integration, etc.) plays only a backgroun role in this book, it is in fact the

actual organizing principle of the course, since the problems of "linear

algebra" can be regarded both as "finite-dimensional projections" and as the

"support" for the basic problems of analysis."

And it goes on to describe the differences between the author's [Shilov's] previous book [An Introduction to the Theory of Linear Spaces (1961 <- that is really old. This book is too. First published in 1977 by Courier Corporation. Hehe. Moving on.)] and this book. I understood up to"LS is entirely

concerned with real spaces, while this book considers spaces over an arbitrary

number field,"

and then could vaguely follow "with the real and

complex spaces being considered as closely related special cases of the general

theory,"

before there came a bunch of terms that hopefully will be defined later and maybe I'll find out that they really mean something actually comprehensible but for now I'll just list them: "Jordan canonical form of the matrix of a linear operator in a real or complex space," "canonical form of the matrix of a normal operator in a complex space equipped with a scalar product," "Hermitian, anti-Hermitian and unitary operators [and their real analogues]" and "infinite-dimensional Hilbert space."Then the author mentions that chapter 11 contains "ancillary material that can be omitted on first reading."

And thanks the editor and an I. Y. Dorfman.

**A definition -- ancillary** Yahoo! Dictionary

an.cil.lar.y (adj) 1. Of secondary importance. 2. Auxiliary.

(n) 1. Something that is subordinate to something else. 2. -Archaic- A servant.

Latin -ancilla- for maidservant, which is a "feminine diminutive of -anculus-" which means servant in "Indo-European roots."

Chapter 1: Determinants

1.1 Number Fields.

1.11 I saw *K* being used as a variable and my eyes glazed over...but I think what the first paragraph is saying is that, like in most of math, linear algebra uses "numbers" (number systems [number fields {any set K of ojects}]). Any set of numbers can be added, subtracted, multiplied, or divided ("subjected to the four arithmetic operations") to make more numbers ("again give elements of K")

Then it lists properties of every pair of numbers A and B in K - commutative, associative, zero, negative element things that take more time to type than it does to memorize -

- for addition (subtraction is a manipulation of addition and negative element)

- for multiplication (division is a manipulation of multiplcation and reciprocal element) *

Natural numbers: 1, 1+1=2, 2+1=3 etc. We assume that none are 0. **

Rational numbers: p/q; p and q are integers and q=/=0

Fields K and K' are isomorphic if

-- Head nod. Must stay awake --

3:01 am

Break time.

Switch to more upbeat radio station. Cut to Doobaba project.

--

3:42 am

Cold. Tired. Not bored, but in the vegetative donotwanttodowork stage.

"Two fields K and K'

are said to be isomorphic if we can set up a one-to-one correspondence between K

and K' such that the number associated with every sum (or product) of numbers in

K is the sum (or product) of the corresponding numbers in K'. The number

associated with every difference (or quotient) of numbers in K will then be the

difference (or quotient) of the corresponding numbers in K'."

I have no idea what that means.Isomorphic=

Two fields. K. K'. [-.-]

K and K' are related.

K and K' = number fields = any set of objects = number systems = group of numbers. Oh.

In each group of numbers... [brain stuttering...failing...fail.]

Sum = +.

Product = x

In K ->

Two numbers: A and B

Sum A and B: A+ B = C

C = number associated with every sum. [I just had an epiphany. {And this is because I am not in a library. (That was probably in bad taste. Sorry.)}]

In K' ->

Two numbers: A' and B'

Sum A' and B': A'+ B' = C'

C = C'

.

...

.

Ugh.

K: A - B = D

K': A' - B' = D'

D = D'

K and K' are isomorphic.

...

I think I'm oversimplifying to the point of wrongness.

--

4:01 am

Search: "isomorphic"

**A definition -- isomorphic** Merriam-Webster

1 a. Being of identical or similar form, shape, or structure.

b. Having sporophytic and gametophytic generations alike in size and shape.

2. related by an isomorphism.

First known use: 1862

** A definition -- isomorphism** ibid.

1: The quality or state of being isomorphic [No. Just no.]

a. Similarity in organisms of different ancestry resulting from convergence.

b. Similarity of crystalline form between chemical compounds.

2: A one-to-one correspondence between two mathematical sets; especially; a homomorphism that is one-to-one - compare ENDOMORPHISM [grrr.]

First known us: circa 1828 [I feel an incongruity here.]

**A definition -- endomorphism** ibid.

A homomorphism that maps a mathematica set into itself - compare ISOMORPHISM.

First known use: 1909

**Wikipedia -- isomorphism**

"In abstract algebra,

an **isomorphism** (Greek: ἴσος *isos* "equal", and μορφή *morphe* "shape") is a bijective map *f* such that both *f*

and its inverse *f* −1 are homomorphisms, i.e., *structure-preserving* mappings. In the more general setting of category

theory, an **isomorphism** is a morphism *f*: *X* → *Y* in a category for which there exists an "inverse" *f* −1: *Y* → *X*, with the property that both *f* −1*f* =

idX and *f f* −1 = idY."

Isomorphism is a morphism. Thank you for informing me. That is probably the only thing I understood.Informally, an

isomorphism is a kind of mapping between objects that shows a relationship

between two properties or operations. *If there exists an isomorphism between*

two structures, we call the two structures *isomorphic**.* In a certain sense, isomorphic

structures are **structurally identical**, if you choose to ignore

finer-grained differences that may arise from how they are

defined.

That explains nothing to me. Right now, I need 1+2=3.Isomorphisms are

studied in mathematics in order to extend insights from one phenomenon to

others: if two objects are isomorphic, then any property which is preserved by

an isomorphism and which is true of one of the objects, is also true of the

other. If an isomorphism can be found from a relatively unknown part of

mathematics into some well studied division of mathematics, where many theorems

are already proved, and many methods are already available to find answers, then

the function can be used to map whole problems out of unfamiliar territory over

to "solid ground" where the problem is easier to understand and work

with.

Better.Consider the group **Z**6, the integers from 0 to 5 with addition modulo 6. Also

consider the group **Z**2 × **Z**3, the ordered pairs

where the *x* coordinates can be 0 or 1, and the y coordinates can be 0,

1, or 2, where addition in the *x*-coordinate is modulo 2 and addition in

the *y*-coordinate is modulo 3. These structures are isomorphic under

addition, if you identify them using the following scheme:

(0,0) →

0

(1,1) →

1

(0,2) →

2

(1,0) →

3

(0,1) →

4

(1,2) →

5

or in general

(*a*,*b*) → ( 3*a* + 4 *b* ) mod 6. For example note

that (1,1) + (1,0) = (0,1) which translates in the other system as 1 + 3 = 4.

Even though these two groups "look" different in that the sets contain different

elements, they are indeed **isomorphic**: their structures are

exactly the same. More generally, the direct product of two cyclic groups **Z***m* and **Z***n* is isomorphic to **Z***mn* if and only if *m* and *n* are

coprime.

...I'll just leave it at that.* Note: in A (B+Y) = AB + AY, for every A, B, Y in K, it is implied that (A+B) Y = Ay + BY.

**

Given two elements N

and E, say, we can construct a field by the rules

N + N=

N,

N + E =

E,

E + E =

N,

N * N =

N,

N * E =

N,

E * E =

E.

Then, in keeping with

our notation, we should write N = 0, E = 1, and hence 2 = 1 + 1 = 0. To exclude

such number systems, we should require that all natural field elements be

nonzero.

End of page 2.4:40 am.

Very cold.

Also paranoid. Saw large spider near lamp about an hour ago. Forgot about spider. Now spider is not in sight.

Shiver.

Definetely cannot go to sleep now.

4:46 am.

Nap?

No. No nap. Daily Show.

5:20 am. Off to read.

9:59 am. I fell asleep.