This new edition is written for first and second year students at both the college or university level. Its real world approach challenges students step-by-step, gradually bringing them to a higher level of understanding from abstract to more general concepts. Real world applications have been added to the new edition, including: Directed graphs Google PageRank Computer graphics Correlation and Variance Finite Fields and Linear Codes In addition to the new applications, the author offers several new exercises and examples throughout each chapter. Some new examples include: motivating matrix multiplication (Chapter 2) a new way to expand a linearly independent set to a basis using an existing basis While some instructors will use the text for one semester, ending at Chapter 5 The Vector Space Rn others will continue with more abstract concepts being introduced. Chapter 5 prepares students for the transition, acting as the "bridging" chapter, allowing challenging concepts like subspaces, spanning, independence and dimension to be assimilated first in the concrete context of Rn. This "bridging" concept eases students into the introduction of vector spaces in Chapter 6.