The activities emphasize a broad range of mathematical thinking. Rather than providing the traditional cycle of Definition-Theorem-Proof, Understanding Linear Algebra aims to develop an appreciation of ideas as arising in response to a need that students perceive. Working much as research mathematicians do, students are asked to consider examples that illustrate the importance of key concepts so that definitions arise as natural labels used to identify these concepts. Again using examples as motivation, students are asked to reason mathematically and explain general phenomena they observe, which are then recorded as theorems and propositions. It is not, however, the intention of this book to develop students' formal proof-writing abilities.