This book addresses two primary deficiencies in the linear systems textbook market: a lack of development of state space methods from the basic principles and a lack of pedagogical focus. The book uses the geometric intuition provided by vector space analysis to develop in a very sequential manner all the essential topics in linear state system theory that a senior or beginning graduate student should know. It does this in an ordered, readable manner, with examples drawn from several areas of engineering. Because it derives state space methods from linear algebra and vector spaces and ties all the topics together with diverse applications, this book is suitable for students from any engineering discipline, not just those with control systems backgrounds and interests. It begins with the mathematical preliminaries of vectors and spaces, then emphasizes the geometric properties of linear operators. It is from this foundation that the studies of stability, controllability and observability, realizations, state feedback, observers, and Kalman filters are derived. There is a direct and simple path from one topic to the next. The book includes both discrete- and continuous-time systems, introducing them in parallel and emphasizing each in appropriate context. Time-varying systems are discussed from generality and completeness, but the emphasis is on time-invariant systems, and only in time-domain; there is no treatment of matrix fraction descriptions or polynomial matrices. Tips for using MATLAB are included in the form of margin notes, which are placed wherever topics with applicable MATLAB commands are introduced. These notes direct the reader to an appendix, where a MATLAB command reference explains command usage. However, an instructor or student who is not interested in MATLAB usage can easily skip these references without interrupting the flow of text.