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I teach honors analysis at an outstanding high school in Palo Alto. We are using some very very old textbooks for the course. The current content is, roughly: 10 weeks on lines and planes and vectors in 2 and 3 dimensions 20 weeks covering Lial/Miller's Finite Math book, including matrices, simplex method for linear programming, Markov chains, game theory, combinatorics, probability, and graph theory 10 weeks of introduction to calculus, using a book on limits and sequences as well as the first few chapters of Foerster's text. What I am looking for is a book to replace at least the first two sections of the course. This book would need to focus on vectors, matrices, lines, and planes, preferably with some applications. If it also included probability and combinatorics, all the better, but we can get those topics elsewhere too. The main problem I see with most texts is that they do not explain vectors and matrices very well. I would like to see the subject matter in the book to talk about the definitions of cross products, matrix products, or determinants. I like the UCSMP Functions, Statistics, and Trigonometry book but it is far too easy for these honors students. What I like about it is that it connects the matrices to rotations and to complex numbers, so it shows why the multiplication ought to be defined the way it is. It also gives some simple applications (like recipes with cost of ingredients) that show the same thing. But it needs some hard problems! Anyway, if anyone has any ideas for a textbook recommendation, I would love to hear them. Internet resources that we could use for discussion of vectors and matrices would be great, too!
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