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Linear Algebra: A Geometric Approach

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One of the most enticing aspects of mathematics, we have found, is the interplay of ideas from seemingly disparate disciplines of the subject. Linear algebra provides a beautiful illustration of this, in that it is by nature both algebraic and geometric.

Our intuition concerning lines and planes in space acquires an algebraic interpretation that then makes sense more generally in higher dimensions. What’s more, in our discussion of the vector space concept, we will see that questions from analysis and differential equations can be approached through linear algebra. Indeed, it is fair to say that linear algebra lies at the foundation of modern mathematics, physics, statistics, and many other disciplines. Linear problems appear in geometry, analysis, and many applied areas. It is this multifaceted aspect of linear algebra that we hope both the instructor and the students will find appealing as they work through this book.

From a pedagogical point of view, linear algebra is an ideal subject for students to learn to think about mathematical concepts and to write rigorous mathematical arguments. One of our goals in writing this text—aside from presenting the standard computational aspects and some interesting applications—is to guide the student in this endeavor. We hope this book will be a thought-provoking introduction to the subject and its myriad applications, one that will be interesting to the science or engineering student but will also help the mathematics student make the transition to more abstract advanced courses.

We have tried to keep the prerequisites for this book to a minimum. Although many of our students will have had a course in multivariable calculus, we do not presuppose any exposure to vectors or vector algebra. We assume only a passing acquaintance with the derivative and integral in Section 6 of Chapter 3 and Section 4 of Chapter 4. Of course, in the discussion of differential equations in Section 3 of Chapter 7, we expect a bit more, including some familiarity with power series, in order for students to understand the matrix exponential.
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