This approach to linear algebra banishes determinants to the end of the work and focuses on the central goal: understanding the structure of linear operators on vector spaces. The author has taken care to motivate concepts and to simplify proofs. Although intended for a second course in linear algebra - following on from matrices and computation - there are no prerequisites other than appropriate mathematical maturity. Thus the book starts by discussing vector spaces, linear independence, span, basics and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite-dimensional spectral theorem. Finally, a variety of exercises in each chapter offer to help students understand and manipulate the objects of linear algebra. This second edition features new chapters on diagonal matrices, linear functionals and adjoints, and on the spectral theorem.