This book consists of the expanded notes from an upper level linear algebra course given some years ago by the author. Each section, or lecture, covers about a week's worth of material and includes a full set of exercises of interest. It should feel like a very readable series of lectures. The notes cover all the basics of linear algebra but from a mature point of view. The author starts by briefly discussing fields and uses those axioms to define and explain vector spaces. Then he carefully explores the relationship between linear transformations and matrices. Determinants are introduced as volume functions and as a way to determine whether vectors are linearly independent. Also included is a full chapter on bilinear forms and a brief chapter on infinite dimensional spaces.The book is very well written, with numerous examples and exercises. It includes proofs and techniques that the author has developed over the years to make the material easier to understand and to compute.Contents: Vector SpacesLinear TransformationsDeterminantsBilinear FormsInfinite Dimensional SpacesReadership: Upper level undergraduate math majors and graduate students.Adjoint Matrix;Axiom of Choice;Basis;Bilinear Form;Cauchy-Schwarz Inequality;Cayley-Hamilton Theorem;Characteristic Polynomial;Companion Matrix;Congruent Matrices;Cramer's Rule;Determinant;Cofactor Expansion;Dimension;Dual Space;Dual Basis;Eigenvalue;Eigenvector;Elementary Matrix;Elementary Row Operation;Field;Gaussian Elimination;Gram-Schmidt Method;Hermitian Bilinear Form;Hilbert Matrix;Image;Jordan Canonical Form;Kernel;Linear Equation;Linear Transformation;Linearly Independent;Matrix; Matrix Addition;Matrix Multiplication;Maximal Linearly Independent Set;Minimal Polynomial;Minimal Spanning Set;Nilpotent Transformation;Nonsingular;Linear Transformation;One-to-One Function;Onto Function;Orthonormal Basis;Parallelogram Law;Quadratic Form;Quotient Space;Rank;Similar Matrices;Reduced Row Echelon Form;Replacement Theorem;Ring of Matrices;Row Echelon Form;Schroeder-Bernstein Theorem;Solution Space;Spanning Set;Square Matrix;Sylvester's Law of Inertia;System of Linear Equations;Trace;Triangle Inequality;Vandermonde Matrix;Vector;Vector Space;Volume Function;Well-Ordering Principle; Zorn's Lemma0Key Features: Each section corresponds to about one week or three one-hour lectures, and includes a set of appropriate exercises Added a brief final chapter on infinite dimensional vector spaces, including the existence of basis and dimension Most undergraduate linear algebra courses are not as sophisticated as this one was. Most math graduate students are assumed to have had a good course in linear algebra, but sadly many have not. Reading these notes might be an appropriate way to fill in the gap
