Saturday, March 26, 2011

Solutions to Insomnia

Solutions to Insomnia

To my mind this title is really 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.]:


"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

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.


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]

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'





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: XY in a category for which there exists an "inverse" f −1: YX, with the property that both f −1f =
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

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


Consider the group Z6, the integers from 0 to 5 with addition modulo 6. Also
consider the group Z2 × Z3, 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) →

(1,1) →

(0,2) →

(1,0) →

(0,1) →

(1,2) →

or in general
(a,b) → ( 3a + 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 Zm and Zn is isomorphic to Zmn if and only if m and n are

...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 + E =

E + E =

N * N =

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

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.


Definetely cannot go to sleep now.

4:46 am.


No. No nap. Daily Show.

5:20 am. Off to read.

9:59 am. I fell asleep.

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