A vector is geometrical; it is an element of a vector space, defined by suitable axioms - whether the scalars be real numbers or elements of a general field. A vector is not an n-tuple of numbers until a coordinate system has been chosen. Any teacher and any text book which starts with the idea that vectors are n-tuples is committing a crime for which the proper punishment is ridicule. The n-tuple idea is not "easier," it is harder; it is not clearer, it is more misleading. By the same token, linear transformations are basic and matrices are their representations.
MacLane, S. (1954). Of course and courses.
Mathematical education is still suffering from the enthusiams which the discovery of this isomorphism [between the ring Hom(V, V) and the ring of all n-by-n matrices with elements in k] 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. [...]
It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out.
Artin, E. (1957). Geometric algebra.
Now let me elaborate on my second criticism, that most linear algebra books confuse the presentation of theory and the technique of calculating via matrices. One way to see that a book makes this mistake is to see that it has a chapter on determinants and matrices before linear transformations are defined. The linear transformation is an easy conceptual thing to talk about and give examples of without matrices. The matrix is a tool for computation, i.e., it is a set of coordinates in a standard array in terms of the bases of U and V. That is, once bases have been selected in U and V so that the points in U and V have coordinates, then a linear transformation from U to V also has coordinates arranged in a rectangular array called a matrix. The matrix, important as it may be for computation, is of no importance in the theoretical or conceptual part of the course nor in the geometric pictures that come along. Presenting matrices before linear transformations and teaching students to work with them is comparable to trying to teach someone to play the piano on a keyboard that isn't attached to any strings. There's no feedback, the student does not see the objective and finds no pleasure in what he's doing. Now I agree that historically matrices came first, vector spaces weren't discovered in their abstract form. For many years a vector space was an R^k for some k and a linear transformation was a transformation given by a system of linear equations represented by the matrix of coefficients. Thus properties of linear transformations had to be formulated as properties of matrices. The abstract point of view, that one could proceed on a different level and work without coordinates, developed during the twenties and thirties. With this new point of view the picture became quite easy and lovely and the theory was disassociated from the mechanism of computation. Thus it is easy to see why the first books on linear algebra had to begin with determinants and matrices, but it seems to me that the conversion to the more recent and simpler view has been much too slow. I don't mind a historical presentation provided it's made clear to the student that matrices are not essential to understanding the theory and that the theory should not be confused with the computations which arise.
Steenrod, N. (1967). The geometric content of freshman and sophomore mathematics courses.
Except for boolean algebra there is no theory more universally employed in mathematics than linear algebra; and there is hardly any theory which is more elementary, in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices. [...]
[...] it is of course possible to translate into matrix language most of the results we have established for linear mappings. We shall not stop to do this: indeed, in practice it is almost always advantageous, when presented with a problem of matrix algebra, to reformulate it in the language, so much more flexible and appropriate, of linear mappings.
Dieudonné, J. (1969). Foundations of Modern Analysis.
In linear algebra, one often sees the determinant of a matrix defined by some ungodly formula, often even with special diagrams and mnemonics given for how to compute it in the 3x3 case, say.
det(A) = some horrible mess of a formula
Even relatively sophisticated people will insist that det(A) is the sum over permutations, etc. with a sign for the parity, etc. Students trapped in this way of thinking do not understand the determinant.
[...]
The larger point here is that although the question asked about having a single wrong definition, really the problem is that a limiting perspective can infect one's entire approach to a subject. Theorems, questions, exercises, examples as well as definitions can be coming from an incorrect view of a subject!
Hamkins, J. D. (2009). answer to What are the most misleading alternate definitions in taught mathematics? in https://mathoverflow.net/a/7952
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The PRESIDENT: I should like to add a few words, in thanking Mr. Carey Francis for his paper. [...] It is true that the Lebesgue integral is very much easier than the Riemann, though naturally the beginnings of it are bound to be a little more difficult. And it is true, in a sense, that the Riemann integral is riddled with awkwardness and exceptions, but when one gets beyond the root of the subject, then the integral of Lebesgue is not really that of generalisation, but of simplification.
Francis, E. C. (1926). Modern Theories of Integration.
[...] the reader will probably observe the conspicuous absence of a time-honored topic in calculus courses, the “Riemann integral.” It may well be suspected that, had it not been for its prestigious name, this would have been dropped long ago, for (with due reverence to Riemann’s genius) it is certainly quite clear to any working mathematician that nowadays such a “theory” has at best the importance of a mildly interesting exercise in the general theory of measure and integration. Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance.
Dieudonné, J. (1969). Foundations of Modern Analysis.
Why do we bother our students with courses on the Riemann integral and Riemann-Stieltjes integrals? [...]
[...] in Stockholm Henstock, knowing the importance of these ideas [his and Kurzweil's and MacShane's], boldly announced that Lebesgue was dead. We think instead that it should be Riemann who is dead, only to be resurrected as Lebesgue by the aid of the trinity Kurzweil-Henstock-MacShane. [...]
What a user needs is the fundamental theorem of calculus and the basic tools such as integration by parts, change of variable and a mean value theorem. The definition should also lead to powerful theorems on the interchange of limit and integration; the lack of these is a serious weakness in the Riemann definition. [...]
It is our contention that the Riemann and Riemann-Stieltjes integrals should become historical curiosities; [...]
Bullen, P. S., & Výborný, R. O. (1990). The teaching of the integral.
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