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Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way.

Again, maybe there are some hidden gems from before the 20th century out there.

Why am I hunting for the oldest sources possible? Well, two very well-known mathematicians who have recently passed (Abhyankar, Voevodsky) and a well-known living physicist (Wolfram) told me to read Grassmann in order to learn linear algebra properly and not be corrupted by the "postmodern turn" in mathematics where the original means and ends of linear algebra have been separated.

I've attempted reading Grassmann's work before, but I would really like a textbook written as close to his lifetime as possible on linear algebra that has exercises.

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    $\begingroup$ What exactly does Grassmann do that isn't properly covered in modern treatments? $\endgroup$ – darij grinberg Oct 28 '18 at 2:40
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    $\begingroup$ Also too broad and unclear what you’re asking. Grassmann is multilinear-algebra rather than linear. “Oldest clone of modern concept” is rarely the right attitude/question. “Original means and ends” of linear algebra are from outside linear algebra (Lagrange’s study of stability under small perturbations, see e.g. Hawkins (2013)). $\endgroup$ – Francois Ziegler Oct 28 '18 at 5:57
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    $\begingroup$ At one point I looked into the origin of the term "linear algebra." It appears to be a 20th-century coinage. Of course, much of the subject matter is very old, but it wasn't called linear algebra. For example there are old books on "matrix theory" but they don't cover exactly the same set of topics that we currently think of as "linear algebra," nor do they think of the subject as being about transformations of vector spaces over a field. So asking for pre-20th century "linear algebra" books may be somewhat anachronistic. $\endgroup$ – Timothy Chow Oct 28 '18 at 18:24
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    $\begingroup$ Google Ngrams suggests that the phrase "linear algebra" originated in 1890 give or take a year but didn't start taking off until about 1920. $\endgroup$ – Noam D. Elkies Nov 8 '18 at 20:40
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    $\begingroup$ @NoamD.Elkies One has to be careful that the phrase first arose in a different sense, in B. Peirce, Linear associative algebra (1870, 1882: §34). In the current sense, Weyl (1919, 1928) is the earliest I’ve seen. $\endgroup$ – Francois Ziegler Nov 9 '18 at 7:50
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The title of the question (oldest textbook) is somehow at odds with its description (textbook closest to Grassmann's life time). Since I understand the motivation of the OP is "to read Grassmann in order to learn linear algebra properly", the obvious textbook would be Peano's "Geometrical Calculus, according to the Ausdehnungslehre of H. Grassmann" (1888).

I quote from C.T. Chong's "Some remarks on the history of linear algebra": "There is no doubt that the extreme clarity of Peano's presentation, in contrast to the notorious difficulty of reading Grassmann's work, helped to spread Grassmann's ideas and made them more popular. This was indeed Peano's objective in publishing the book, as he stated in the forword."

The original Italian is here, the English translation is published by Springer. Every chapter has problems with solutions, which you could call "exercises" I think.

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The Nine Chapters on the Mathematical Art dates from between the 10th-2nd centuries BC.

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    $\begingroup$ I don't think this is a good question, but the (independent) ancient Chinese origins of linear algebra deserve to be better known. $\endgroup$ – David Roberts Oct 28 '18 at 10:36

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