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I'm quite desperate to understand the historical motivation and origin of all of the "geometrical" concepts of linear algebra, namely:

  • The concept of thinking of elements of $\mathbb R^n$ or some other vector space as points in a "space", and of subspaces as lines and planes.
  • The notions of norm and inner product as generalizations of length and angle.

More generally, I'm interested in any detailed histories of linear algebra, although my main motivation is still to try and get over my intense phobia of norms and inner products. I found the book The Genesis of the Abstract Group Concept very helpful with similar troubles about the motivations of group theory, but I can't find anything similar for linear algebra, and brief, superficial synopses on Wikipedia articles just aren't cutting it.

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Are you familiar with Michael J. Crowe's book, A History of Vector Analysis? While I haven't read the book this article is well worth a read, and it seems to be a good summary.

Of course, vector analysis is the precursor to linear algebra, so it won't directly address your question. Crowe does discuss briefly Grassmann's Ausdehnungslehre, one of the roots of (n-dimensional) linear algebra, and the (somewhat convoluted) history of the inner product.

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  • $\begingroup$ This book looks very like the kind of thing I'm interested in. A shame it doesn't explicitly cover vector spaces, but it's close. $\endgroup$ – Jack M Nov 7 '14 at 0:17
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This is specifically about the history of linear algebra, history of Matrices and determinants.

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From your sentence "my main motivation is still to try and get over my intense phobia of norms and inner products" I conclude that you need first of all a good book in Linear algebra itself, rather then history of linear algebra. In English, I recommend the textbook of P. Lax. There is a nice book of Dieudonne Algèbre linéaire et géométrie élémentaire (there is an English translation) which gives an exposition of high school geometry from the linear algebra point of view. Essentially this is the book which does all linear algebra in dimensions 2 and 3. That is elementary geometry, only exposed in a modern way.

On the history of linear algebra there is another book of Dieudonne, Abrege d'histoire des mathematiques, vol. I which explains genesis of these notions.

But I have to repeat that genesis was quite complicated and convoluted, before the modern clarity and simplicity were reached. So in this particular case, I recommend NOT to follow the historical development if your problem is to understand the linear algebra itself. Only AFTER you overcome your "fobia of norms and inner products" you may read some of this history with a profit.

EDIT. Another good book is MR1885576 Givental, Alexander Linear algebra and differential equations. Berkeley Mathematics Lecture Notes, 11. American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 2001.

It teaches you linear algebra in dimension 2. That is the linear algebra part covers the SAME material as middle school geometry course. Only in the modern language. If you had a geometry course at school, there must be nothing unfamiliar to you in linear algebra in dimension 2.

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  • $\begingroup$ While I appreciate the reference, Lax's textbook, like many others, simply introduces the definition of the Euclidean norm, points out that it generalizes something familar, and then just assumes the student will find that natural. This is the typical modern approach, and while this is subjective I really don't find it natural to simply generalize because we can. Thus I've been looking for historical background. I did check out Dieudonné's history from the library, though. $\endgroup$ – Jack M Nov 6 '14 at 20:22
  • $\begingroup$ Euclidean norm indeed generalizes something familiar: this is the length of a vector in the plane. If the notion of length is not familiar, then probably one has to start with elementary geometry, not linear algebra. Lax's book is outstanding because it gives many examples of applications. $\endgroup$ – Alexandre Eremenko Nov 6 '14 at 20:33
  • $\begingroup$ And this is not a generalization for the sake of generalization: it is a USEFUL generalization, and a good linear algebra book must show this. On my opinion, Lax does. But there are also other good books, no doubt. $\endgroup$ – Alexandre Eremenko Nov 6 '14 at 20:35
  • $\begingroup$ @JackM : you say "I really don't find it natural to simply generalize because we can". Some people might say that it is one of the main driving forces behind mathematics. But on your specific example, distance, before generalization to n-space, there is experiment, exploration and wonder, art, controversies. You can use the same concept of distance on a line and on a plane, and you don't have to invent something new for space. The very idea that our space has 3 dimensions is an amazing conceptual effort. The ideas of parameter, variable, coordinate and much more were partly born of all this. $\endgroup$ – ogerard Mar 25 '15 at 10:32
  • $\begingroup$ Is there a translation of Dieudonne, Abrege d'histoire des mathematiques, vol. I ? I tried to Google it, but I have trouble seeing through the French. $\endgroup$ – James S. Cook Nov 3 '18 at 18:11
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The 3-volume book by Thomas Muir The Theory of Determinants in the Historical Order of Development covers a narrower subject, but its first sections are very interesting to understand the early history of linear algebra. The notion of determinant predates other notions of linear algebra, and the book gives an exhaustive listing of all its early occurrences from Leibniz onwards.

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