Socrates J.T. A Portrait of Linear Algebra

3d edition. — Pasadena: Kendall Hunt Publishing Company, 2016. — 885 p.A Portrait of Linear Algebra takes a unique approach in developing and introducing the core concepts of this subject. It begins with a thorough introduction of the field properties for real numbers and uses them to guide the student through simple proof exercises. From here, we introduce the Euclidean spaces and see that many of the field properties for the real numbers naturally extend to the properties of vector arithmetic. The core concepts of linear combinations, spans of sets of vectors, linear independence, subspaces, basis and dimension, are introduced in the first chapter and constantly referenced and reinforced throughout the book. This early introduction enables the student to retain these concepts better and to apply them to deeper ideas.
The Four Fundamental Matrix Spaces are encountered at the end of the first Chapter, and transitions naturally into the second Chapter, where we study linear transformations and their standard matrices. The kernel and range of these transformations tells us if our transformations are one-to-one or onto. When they are both, we learn how to find the inverse transformation. We also see that some geometric operations of vectors in 2 or 3 are examples of linear operators.
Once these core concepts are firmly established, they can be naturally extended to create abstract vector spaces, the most important examples of which are function spaces, polynomial spaces, and matrix spaces. Linear transformations on finite dimensional vector spaces can again be coded using matrices by finding coordinates for our vectors with respect to a basis. Everything we encountered in the first two chapters can now be naturally generalized.
One of the unique features of this book is the use of projections and reflections in 3, with respect to either a line or a plane, in order to motivate some concepts or constructions. We use them to explore the core concepts of the standard matrix of a linear transformation, the matrix of a transformation with respect to a non-standard basis, and the change of basis matrix. In the case of reflection operators, we see them as motivation for the inverse of a matrix, and as an example of an orthogonal matrix. Projection matrices, on the other hand, are good examples of idempotent matrices.
The second half of the book goes into the study of determinants, eigentheory, inner product spaces, complex vector spaces, the Spectral Theorems, and the material necessary to understand and prove the Fundamental Theorem of Linear Algebra, and its twin, the Singular Value Decomposition. We also see several applications of Linear Algebra in science, engineering, and other areas of mathematics. Throughout the book, we emphasize clear and precise definitions and proofs of Theorems, constantly encouraging the student to read and understand proofs, and to practice writing their own proofs.