

Summary of MIT's Linear Algebra. Lecture 3: Matrix Multiplication and Inverses - pkrumins
http://www.catonmat.net/blog/mit-linear-algebra-part-three/

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btilly
That is an approach to linear algebra that I dislike.

"Here is a magic formula for multiplying matrices. Here is what you can do
with it. Here are some examples." _Without any clue WHY you would want to!_

The approach I like is to discuss what a linear function is, show how you can
represent linear functions by matrices (given two finite dimensional vector
spaces and bases for those vector spaces, there is a 1 to 1 correspondence
between linear functions and matrices), then show how function composition
turns into matrix multiplication. The common argument against this approach is
that it demands too much from the students. However I disagree. To me it
motivates the subject, results in fewer magic formulas, and it makes it easier
later for people to see how to apply it in practice.

For instance take the associative law of matrix multiplication. It is
straightforward that:

    
    
      F o (G o H) (x) = F((G o H)(x)) = F(G(H(x)))
      (F o G) o H (x) = (F o G) (H(x)) = F(G(H(x)))
    

So function composition is associative. Since matrix multiplication is a
representation of function composition, it must be associative as well.

Now try to write out a direct proof that matrix multiplication is associative.
It will be much longer, much harder, and at the end you'll have no clue how
someone could have ever thought up this definition or noticed that THIS
definition satisfied the associative law.

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madars
There are two introductionary books on linear algebra I would recommend.
"Linear Algebra Done Right" by Axler gives you a _very_ nice and rigorous way
to the foundations (this text requires a small bit of mathematical maturity,
so be aware).

After reading Axler's book you should go and grab a copy of Strang's
"Introduction to Linear Algebra" to view the applied side of linear algebra
(this is the book mentioned in the article).

Strang's book is a terrible choice for first textbook of linear algebra (for
the reasons btilly laid out), but it presents very imortant view on linear
algebra, provided that you have mastered the foundations enough to compensate
for the lack of rigour.

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timtadh
Really why by a book when you can just grab a PDF. I used Linear Algebra Done
Wrong for my class this semester, and found it a pretty good book for being
free. check it out: www.math.brown.edu/~treil/papers/LADW/LADW.pdf

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jimbokun
Skimming the preface, this book appears to be something like the approach
btilly describes. Thanks for the link. Intend to download to my iPod Touch to
compete for my rare moments of free time with Elements of Statistical
Learning, and the other free Math and CS books I'm sure will be on there soon.

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jchonphoenix
I don't understand what's so interesting about MIT's linear algebra course...
It seems pretty standard. Penn State and Pitt's cover the exact same course
material.

And theirs is pretty equivalent to CMU's Matrix Algebra (which covers a bit
more than MIT)

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catzaa
> I don't understand what's so interesting about MIT's linear algebra course

The lectures are free to download (which is awesome to 3rd world folks like me
:) )

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jmtame
gets fun once you start going into vector spaces..

