Re: good text for linear algebra?

In article <d4qu1d$pe1$1@xxxxxxxxxxxxxxxxx>,
Dave Rusin <rusin@xxxxxxxxxxxxxxxxxxxxx> wrote:
>In article <f14f9$42708b36$d468cfd2$12701@xxxxxxxxxxxxxxxxxxxx>,
>Willem H. de Boer <wdeboer@xxxxxxxxxxxxxxxxxx> wrote:

>>Definitely Sheldon Axler's book. His approach is to teach linear algebra
>>via linear transformations on vector spaces; it's not until the later
>>chapters
>>that he introduces matrices (as representations of linear transformations
>>after a choice of basis). Also, the chapter on eigenvalues comes before
>>his chapter on determinants!

>The fact that you think an exclamation mark belongs here is symptomatic
>of the prevailing mentality that Axler wants to overcome.
>Eigenvalues (and singular values for that matter) are much more
>central to the ideas of linear algebra than the existence of a
>multiplicative map M_n(R) --> R .

>My experience with determinants and eigenvalues is akin to my experience
>with integrals and antiderivatives. We all go through Riemann sums and
>whatnot to explain what integrals _are_. Five minutes after having
>shown students the Fundamental Theorem of Calculus, none of them can
>distinguish integrals from antiderivatives any more, and they will
>tell you that e.g. \int_R exp(-x^2) dx "can't be done". So too with
>eigenvalues. Sample question: ask them to compute the eigenvalues of a
>diagonal matrix (after they've seen that eigenvalues _may_ be computed as
>roots of det(A-XI) ). Do you really think they understand what
>eigenvalues _are_ any more?

This suggests that Axler's approach is doomed, despite his
laudable intentions, since eventually the students will get to that
chapter on determinants.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada







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