From: Jim Choate <ravage@ssz.com>
Date: Mon, 27 Oct 1997 13:41:21 +0800
To: cypherpunks@ssz.com (Cypherpunks Distributed Remailer)
Subject: Re: Orthogonal (fwd)
Message-ID: <199710270528.XAA02347@einstein.ssz.com>
MIME-Version: 1.0
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Forwarded message:

> Date: Sun, 26 Oct 1997 19:13:12 -0800
> From: Kent Crispin <kent@bywater.songbird.com>
> Subject: Re: Orthogonal (fwd)

> > > I do believe the use of the term this way was inspired by the 
> > > notion of a 'basis' in a vector space -- a set of orthogonal
> > > vectors that span the space, ideally, unit vectors.
> > 
> > Can you better define the term 'basis'?
> 
> This is basic linear algebra:
> 
> V a vector space -- the set of all (s1,s2,s3,...,sn), where si is an
> element of the set of reals.

Actualy si (scalars) can be rational, real, or complex. It is also possible 
to use more general structures such as fields. [1]

Note that complex numbers numbers may be used, in other words a vector can
be used to multiply another vector.

> A set of vectors {v1,v2,...,vm} in V is

A vector of vectors, and you haven't even really defined vector yet...:(

> linearly independent if there is no set of scalars {c1,c2,...,cm} with
> at least one non-zero element such that sum(ci*vi) == 0

Are you saying:

(c1*v1)+(c2*v2)+...+(cn*vn) <> 0

where at least one ci is <>0, is some sort of test for membership in a vector
space? What about,

(1/c1*v1)+(-1/c2*v2)+(0*v3)...+(0*vn) = 0,
     1         -1

if so it should be clear that except for the case of c=0 there is always a
way to take two, which is clearly more than one, of the scalars and cause
the sum to be zero. Now if you have a single non-zero scalar multiplier then
it seems reasonable that you can create such a structure that is always <>0.

> A set of
> vectors S spans

In other words 'are expressible in'?

> a vector space V iff every element of V can be expressed
> as a linear combination of the elements of S.
              ^^^^^^^^^^^
              Need to better define this one.

The operations that are applicable to a vector space are: [1]

1.  Associative law of addition: (x+y)+z = x+(y+z)
2.  Commutative law of addition: x+y=y+x
3.  Existance of zero: x+0=x for all x in V
4.  Existance of inverses: x+(-x)=0
5.  Associative law of multiplication: a(bx)=(ab)x
6.  Unital law: 1*x=x
7.  First distributive law: a(x+y)=(a*x)+(a*y)
8.  Second distributive law: (a+b)*x=(a*x)+(b*x)

where x is of the form 'asubn(x^n)+bsubn-1(x^n-1)+...+asub1x+asub0'

This is a remarkably circular definition to present. In clearer wording;

The set of vectors S is expressible in a vector space V iff S is contained in 
V.

While this may be a requirement or test for membership in a vector space (as
used below) it doesn't qualify as a definition.

> Finally, a basis for 
> V is a linearly independent set of vectors in V that spans V.  A 
> space is finite dimensioned if it has a finite set for a basis.  The 
                                                 ^^^
                                                 set of what?

> standard basis (or natural basis) for a vector space of dimension n 
> is th set of vectors
> 
> (1,0,0,...0)
> (0,1,0,...0)
> (0,0,1,...0)

In other words, any 'vector' can be expressed as the sum of the
multiplication of unit elements by some set of scalars.

Is this definition something you just wrote or would you be so kind as to
give the reference if it isn't.

This definition uses 'basis', doesn't define it. You can't use a term to
define the term, it's called circular reasoning. My question still stands,
can you please better define 'basis'. Or is your claim that basis is simply
a way of stating 'elements of unit magnitude'?

Furthermore, a definition of vector space has nothing to do with orthogonal
measurement systems, it does have to do with polynomials. It just so
happens that this sort of geometry shares the same sort of rules, that
doesn't make them the same thing. There is nothing in your description, in
particular the nifty little (*,*,*,...*)'s that implies any sort of
measurement system based on line-segment axis that are 90 degrees apart,
merely that there is more than one variable (ie x,y,z,...a) involved. 

Vector space -  a vector space is a set of objects or elements that can be
                added together and multiplied by numbers (the result being
                an element of the set), in such a way that the usual rules
                of calculations hold. [1]


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[1]  The VNR Concise Encyclopedia of Mathematics
     Gellert, Kustner, Hellwich, Kastner
     ISBN 0-442-22646-2
     17.3 Vector spaces
     pp. 362


[Hallowen Trivia]

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   to Vlad Tepish, whose name means 'son of the devil', he was further known
   as Vlad the Impaler because he would impale people on poles and place them
   along the roads in Transylvania. When he died in 1496 his head was
   impaled on a stick as well.