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From: laura@hoptoad.uucp (Laura Creighton)
Newsgroups: net.cse
Subject: Re: Re: Math and CS
Message-ID: <609@hoptoad.uucp>
Date: Tue, 11-Mar-86 16:24:54 EST
Article-I.D.: hoptoad.609
Posted: Tue Mar 11 16:24:54 1986
Date-Received: Fri, 14-Mar-86 04:30:54 EST
References: <256@hropus.UUCP> <6400005@ccvaxa> <77@umcp-cs.UUCP> <1194@mit-eddie.MIT.EDU> <587@hoptoad.uucp> <5157@stolaf.UUCP>
Reply-To: laura@hoptoad.UUCP (Laura Creighton)
Organization: Nebula Consultants in San Francisco
Lines: 69

Linear Algebra -- boy is there a lot of difference in how the courses are
presented in different places...

I got a lot of mail about this subject -- and I am still gettting it.
Thanks for the mail, people.  But from it I can see that what a lot
of people took in linear algebra was matrices, and calculating determinents,
and Eigenvalues and so on.  Where I come from, matrtic manipulation and
determinents are taught in high school.  And I learned about Eigenvalues
in physics courses.  And I never had any problem with them -- indeed, one
of the most enjoyable things I have done in recent times is read the
microcode which went into an array processor which did this sort of thing.
(i found 2 bugs, too!).

But I had no idea that this had anythign to do with linear algebra. I got
a copy of Schaum's Outline for linear algebra, though and so I can see
that it must be..but this wasn't the course 

What we got was  pure theory.  There were no numbers, except as subscripts
and superscripts.  There were definitional questions, such as:

	What is a vector space?
	What is a sub-space?  bases and dimensions? fields?

And a literal hundred lemmas and theories -- which is what I couldn't
memorize.

I found an example in some old papers I have.  On the right hand side of
the page are notes to myself on how to convert a pdp-11 running PWB to
v7.  And on the left side...

	Let V be a vector space and suppose T and U are linear operators
on V such that
	(a)	U is onto
	(b)	The Null spaces of T and U are finite-dimensional.
Then the null space of TU is finite-dimensional and 

	dim(N(TU)) = dim(N(T)) + dim(N(U)).

proof:	let p = dim(N(T)), q = dim(N(U)), and {u1,u2,...,up} and {v1,v2,...vq}
	be bases for N(T) and N(U) respectively.  SInce U is onto, we can
	choose for each i (i =1,...,p) an element wi (- V such that
	U(wi) = ui.  Thus we obtain a set of p elements {w1,w2...,wp}.
	Note that for any i and j, wi != vj, for otherwise 
	ui=U(wi)=U(vj)=0-a contradiction.

[I wrote it that way.  I think now that it should be
	...=U(vj)=0	-a contradiction!]
	
	Hence the set

		B={w1,w2,...wp,vi....,vq} contains p+q distinct elements.

	To prove the lemma it suffices to show that B is a basis for
N(TU).

-------------

It goes on and on in that vein for another 40 lines.
I had page after page after page of this stuff to memorize.  And you know
something?  I still don't know what a vector space *is*...  It is still
all so many words.

I think that if people want to discuss the beauty of linear algebra with
me we should do this in mail or in net.math.  But  I am curious -- is
this what other people had in linear algebra?  Or number crunching?
-- 
Laura Creighton		
ihnp4!hoptoad!laura  utzoo!hoptoad!laura  sun!hoptoad!laura
toad@lll-crg.arpa