The title of the question (oldest textbook) is somehow at odds with the following 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.

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.