Text identical to archival version, embedded hyperlinks activated.

                   October 30, 1997

TO: Dr. Carl C. Cowen
FR: Kirby Urner, 4D Solutions
RE: Teaching linear algebra and computing

Greetings Carl --

Re your: http://www.maa.org/features/cowen.html

I was just exploring the MAA web site again and come across
your article re teaching linear algebra and the importance
of computing in this connection.

I thought I'd share with you my recent experiences as a 
curriculum writer integrating some aspects of linear algebra,
the computer package MathCad, a programming language (xBase),
a free rendering program (Povray), and HTML, to create some 
interesting (I think anyway) web pages accessible to advanced 
high school or college students.

You can see the results at:

What I like about MathCad is it uses standard text book
typography (with a few idiosyncracies), meaning I can test
some of my equations and matrices as live objects, and 
then capture screen fragments as GIFs for embedding in 
HTML.  This gives me access to the large radical signs
over compound expressions etc. without having to get into
LaTex or PostScript or anything.

My papers also share snippets of computer language and
point to full source code on other pages.  Although using
xBase is somewhat unusual in this context, I find it 
congenial -- the syntax is object oriented these days,
plus we have a lot of native power when it comes to 
tabular data.  But really, the choice of language is 
not that central and I make a point on my web pages of
referencing other languages (APL, Logo, Java).

I think it's important to understand that matrix algebra
does owe a lot to typography, the ability to typeset 
rectangular arrays and lots of subscripts and superscripts.  
And computers come in so naturally here because programmers 
use the exact same concepts of stored values indexed by row 
and column.  A 'matrix' is what a programmer calls an 'array' 
after all -- at least one linear algebra books in my possession
makes it clear that the generic 'matrix' is any row and column 
storage system, with those matrices used in eigenvalue 
computations etc. being a special case of a more generic 

Finally, the links between matrix algebra and geometry are what 
bring the ray tracing into play.  The xBase 'turtles' (concept 
from Logo) know how to write out data in a form which Povray 
can process into brightly colored spatial renderings -- suddenly 
students have access to all the tools they need to make web pages 
that compete with million dollar text books in terms of graphical 

If you have the time to give my papers a more than cursory 
inspection, you may find some problematic aspects vis-a-vis
formal linear algebra concepts.  Like, I use matrix notation
to define conversion between two coordinate systems, except
one is not xyz, yet maps volume, yet uses 4-tuples, yet uses 
no negative numbers.  This may take me outside the domain of 
formal linear algebra into more unexplored territory, but that 
need not distract nor detract from the points I'm making above, 
re the opportunities our students now have to explore, program 
and document their mathematical explorations by exploiting the 
many positive synergies offered by recombining newer and older 



written quite a bit earlier, bring in VRML, something 
I hope to get back to more.

PPS:  I should add that, in addition to the above resources,
the professional appearance of some of these web papers 
owes considerably to Macmillan Publishing (although not in
a legal sense as the copyright on the images in question 
is no longer a Macmillan property).

Synergetics on the Web
maintained by Kirby Urner