You will probably find the beginning of the book to be largely review. You may have seen much of Chapters 2 (Vector spaces) and 3 (Linear transformations) and some of Chapter 4 (Eigenvalues) in a first linear algebra course, but I do not assume that you have mastered these concepts. As befits review material, most of the concepts are presented from the beginning, but quickly. If you thoroughly understood your first course, you should be able to skim these chapters, concentrating on the last section or two of each.

I typically present each major concept in three settings. The first setting is in R^n, where the problem is essentially a (frequently familiar) matrix computation. The second setting is in a general n-dimensional vector space, where a choice of basis reduces the problem to one on R^n. The key is to choose the right basis, and I put considerable emphasis on understanding what stays the same and what changes when you change basis. The third setting is in an infinite dimensional vector space. The goal is not to develop a general theory, but for you to see enough examples to start building up intuition.

Infinite dimensional spaces appear more and more often later in the book. While (almost) all results in finite dimensions are proven, most infinite dimensional theorems (such as the spectral theorem for bounded self-adjoint operators on a Hilbert space) are merely stated, and in some cases I just argue formally, by analogy to finite dimensions. Although such analogies can sometimes fail (I give examples), they are a very good intuitive starting point.

This book is aimed at a mixture of math, physics, computer science, engineering and economics majors. The only absolute prerequisite is familiarity with matrix manipulation (Gaussian elimination, inverses, determinants, etc.). These topics are invariably covered in a first linear algebra course, but can also be picked up elsewhere. Basic vector space concepts are covered from the beginning in Chapters 2 and 3. However, if you have never seen a vector space (not even R^n), you may have difficulty keeping up. These chapters are aimed primarily at students who once were exposed to vector spaces but may be rusty. Beyond Chapter 3, no prior knowledge is expected or required.