Elements of Linear Algebra, Differential & Integral Calculus pdf

Two subjects. One course. One book.

Linear algebra and calculus are usually taught from two books, in two notations, with two habits of mind. This one develops them as a single course, so the tangent line you meet in Chapter 10 is recognisably the linear map you met in Chapter 4.

No previous knowledge of calculus is assumed. A solid secondary-school background is all you need — Chapter 0 covers every prerequisite before the course begins. From there, twenty-one chapters run from vector spaces, matrices and eigenvalues through limits, derivatives, integrals and infinite series.

It is written for first-year students of engineering and the sciences: the ones who meet mathematics as a tool of their own discipline and need to understand it, not just pass it. The emphasis falls throughout on conceptual understanding, on geometric intuition, and on solving problems. It is the fruit of more than twenty years of teaching first-year mathematics to engineers.

Every chapter is linked to free companion material. Scan a code with your phone and you get the thing the printed page cannot give you:

• Animations — move the parameter, watch the theorem move.

• Self-assessment quizzes — one per chapter, answered and explained.

• AI proof walkthroughs — each theorem opens with the prompt ready, so you can interrogate the proof line by line.

• Maxima and Python activities — runnable, with the commands given

• Fully worked, step-by-step examples — every step shown, none assumed.

• Unsolved exercises of graded difficulty, so you can find your level and climb.

• Applications drawn from engineering and the sciences.

• Historical and biographical notes, from Newton and Leibniz to Cauchy and Riemann, beside the mathematics they created.

• A chapter summary and review exercises at the end of every chapter.

Part I — Elements of Linear Algebra: linear vector spaces; matrix algebra and determinants; linear systems; linear transformations; eigenvalues and diagonalization; discrete dynamical systems.

Part II — Differential and Integral Calculus: functions, limits and continuity; the derivative and its applications; monotonicity and curve sketching; indefinite and definite integrals; techniques and applications of integration; improper integrals; sequences and infinite series.

Nikolaos M. Matzakos is Associate Professor of Applied Mathematics at ASPETE, Athens, and holds a PhD in Mathematics from the National Technical University of Athens. He has taught first-year mathematics to engineering students for over twenty years, and his research spans mathematics education and the use of artificial intelligence in the teaching of mathematics.