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.
Best Answer
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.