Re: Linear Algebra Done Right

On 31 Dez. 2008, 20:56, pwoly...@xxxxxxxxx wrote:
Hi,

I really enjoyed Sheldon Axler's book "Linear Algebra Done Right". One
thing about the book that is a little bit of a drawback is his use of
lists of vectors, rather than just considering them as sets. Axler is
able to use lists in such a way that he can prove some results quite
nicely. My problem with his approach however is that it seems somewhat
cumbersome for students to use in writing formal proofs, and Axler
never seems to develop a good notation to complement/support it. I am
curious if anyone has ever seen an approach similar to that presented
by Axler that is a bit cleaner and easier to use in writing formal
proofs?

Thanks,

Peter

I assume the lists play a role especially with linear dependency and
bases etc.
Doing that with sets instead of lists (aka. families) of vectors would
be
the wrong thing to do.
For example:
The set(!) of R^2 vectors {(1,0), (t,0)} is linear dependant only if
t != 1.
If t=1 it is a singleton set containing a nonzero vector and hence
lin. independant!
The corresponding list always has two w'members, eve for t=1, and
hence it
is always lin. dependant.
Similarly, the column vectors of
( 1 2 1 )
( 4 1 4 )
form a basis of R^2 (when using sets instead of lists)
whereas the column vectors of
( 1 2 1 )
( 4 1 3 )
don't.
.

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