By J.-L. Dorier
This ebook offers the cutting-edge learn at the instructing and studying of linear algebra within the first yr of collage, in a world viewpoint. It goals at giving to school academics in control of linear algebra classes a variety of details from works together with theoretical and experimental concerns. those works try and higher comprehend the that means of linear algebra in an epistemological process, in addition to the limitations and the problems in its educating and studying. in addition they current educating designs with the research in their experimentation. besides the fact that, it'd be useless to anticipate a astonishing method for educating linear algebra, fairly parts of mirrored image are given to the reader with keys for their personal experimentation. in the course of the number of the works offered, this publication additionally discusses theoretical concerns that are constant for all study in maths schooling at tertiary point.
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Additional info for On the Teaching of Linear Algebra, 1st Edition
Mais cette nouvelle caracteristique suivant des figures de vue, ne peut manquer de donner en meme temps la solution et la construction et la demonstration géometrique, le tout d'une maniere naturelle et par une analyse. , 1:384) The geometrical analysis presented by Leibniz was based on a relation of congruence between n-tuples of points: two couples of points are congruent if their two points are at an equal distance from each other, two triplets are congruent if the two triangles they determine can be brought into coincidence, etc.
Yet, Grassmann does not explicitly discuss this point either in §20 or further on, although it is implicitly admitted. In spite of these imperfections, Grassmann's use of what will be known later as the exchange method is fundamental. Indeed, it is one of the ways of proving the invariance of the number of elements in any basis of a vector space, which does not make use of coordinates and is therefore the most intrinsic. We will see that none of Grassmann's direct followers made reference to this part of his work and therefore had a less elaborate presentation of the concept of basis.
The tools were more sophisticated, but their use required such technique that intuition was averted. 4. Georg Ferdinand Frobenius Frobenius was one of the first (if not the first) to have introduced a crucial change in the approach of linear problems. In a text from 1875 entitled, Über das Pfaffsche Probem, he defines, in modern terms, the notions of linear dependence and independence simultaneously for equations and n-tuples, making the link with the conception of inclusive dependence quite clear: Mehrere particuläre Lösungen χ Α1(χ), .