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Required fields*

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    $\begingroup$ "Mathematical education is still suffering from the enthusiams which the discovery of this isomorphism has aroused. The result has been that geometry was eliminated and replaced by computations. Instead of the intuitive maps of a space preserving addition and multiplication by scalars (these maps have an immediate geometric meaning), matrices have been introduced. From the innumerable absurdities -from a pedagogical point of view - let me point out one example and contrast it with the direct description: $\endgroup$ Commented Jun 19, 2010 at 10:25
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    $\begingroup$ Matrix method: A product of a matrix $A$ and a vector $X$ (which is then an n-tuple of numbers) is defined; it is also a vector. Now the poor student has to swallow the following definition: A vector X is called an eigenvector if a number $\lambda$ exists such that $$AX=\lambda X.$$ Going through the formalism, the characteristic equation, one then ends up with theorems like: If a matrix A has n distinct eigenvalues, then a matrix $D$ can be found such that $DAD^{-1}$ is a diagonal matrix. The student will of course learn all this since he will fail the course if he does not. $\endgroup$ Commented Jun 19, 2010 at 10:31
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    $\begingroup$ Instead one should argue like this: Given a linear transformation f of the space $V$ into itself. Does there exist a line which is kept fixed by $f$? In order to include the eigenvalue $0$ one should then modify the question by asking whether a line is mapped into itself. This means of course for a vector spanning the line that $$f(X)=\lambda X.$$ Having thus motivated the problem, the matrix A describing f will enter only for a moment for the actual computation of X. It should disappear again. $\endgroup$ Commented Jun 19, 2010 at 10:36
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    $\begingroup$ Artin's view here is very much the view of Dieudonné (as expressed in his book on linear algebra). I think that Arnold simplifies the world into black and white and attacks a straw boogeyman named Bourbaki. Hoffman and Kunze gives a very nice account of these geometric aspects as well as the algebraic ones. $\endgroup$ Commented Jun 19, 2010 at 10:43
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    $\begingroup$ Harry, don't get all worked up. Let me repeat the key point: $\textit{It's a harmful fallacy that conceptual understanding and applications are mutually exclusive.}$ There isn't any application in sight in your quotes, just comparisons between different formalisms. And for what it's worth, the last one is perfectly in line with Arnold's philosophy, while it doesn't conform well to Bourbaki's way of thinking. $\endgroup$ Commented Jun 20, 2010 at 1:06