A Portrait of Linear Algebra

Linear Algebra is a powerful tool that allows scientists to model physical phenomena and efficiently solve the systems of equations that arise from them. It is essential to almost every field of mathematics, such as geometry, topology, number theory, and abstract algebra. As such, it is not only computationally useful, with applications in data science, quantum mechanics, statistics, computer science, and all branches of engineering, but it also a great opportunity for the student to explore the beauty and elegance of mathematics. A Portrait of Linear Algebra introduces the students to the algorithms and structures of this subject with rigorous definitions, clear explanations, and interesting examples. The exercises include both computational as well as theoretical problems that will challenge the student's understanding. You will not just see how Linear Algebra works, but more importantly why it works. The structures of Linear Algebra vector spaces, matrices, linear transformations, and their interrelated properties are built from the field axioms of the real number system. This enables us to prove almost all the Theorems discussed in the text, with many exercises that allow the student to develop their own proof-writing skills. A Portrait of Linear Algebra rigorously prepares the student to tackle more advanced techniques and applications of Linear Algebra, in whatever field of interest they may be. There are also many projects that will direct the student to go deeper into the subject, such as using projections to draw a three-dimensional object, rotating a vector in space around an arbitrary axis, and simultaneously diagonalizing commuting matrices.