We define composition of linear transformations, inverse of a linear transformation, and discuss existence and uniqueness of inverses.https://ximera.osu.edu/la/LinearAlgebra/LTR-M-0030/main

We introduce matrices, define matrix addition and scalar multiplication, and prove properties of those operations.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0010/main

We introduce matrix-vector and matrix-matrix multiplication, and interpret matrix-vector multiplication as linear combination of the columns of the matrix.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0020/main

We define the transpose of a matrix and state several properties of the transpose. We introduce symmetric, skew symmetric and diagonal matrices.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0025/main

We interpret linear systems as matrix equations and as equations involving linear combinations of vectors. We define singular and nonsingular matrices.https://ximera.osu.edu/la/LinearAlgebra/MAT-M-0030/main

We solve systems of equations in two and three variables and interpret the results geometrically.https://ximera.osu.edu/la/LinearAlgebra/SYS-M-0010/main

We introduce the augmented matrix notation and solve linear system by carrying augmented matrices to row-echelon or reduced row-echelon form.https://ximera.osu.edu/la/LinearAlgebra/SYS-M-0020/main

We introduce Gaussian elimination and Gauss-Jordan elimination algorithms, and define the rank of a matrix.https://ximera.osu.edu/la/LinearAlgebra/SYS-M-0030/main

We define a homogeneous linear system and express a solution to a system of equations as a sum of a particular solution and the general solution to the associated homogeneous system.https://ximera.osu.edu/la/LinearAlgebra/SYS-M-0050/main

We define closure under addition and scalar multiplication, and we demonstrate how to determine whether a subset of vectors in R^n is a subspace of R^n.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0020/main

We define the row space, the column space, and the null space of a matrix, and we prove the Rank-Nullity Theorem.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0040/main

We state the definition of an abstract vector space, and learn how to determine if a given set with two operations is a vector space. We define a subspace of a vector space and state the subspace test. We find linear combinations and span of elements of a vector space.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0050/main

We revisit the definitions of linear independence, bases, and dimension in the context of abstract vector spaces.https://ximera.osu.edu/la/LinearAlgebra/VSP-M-0060/main

This is a book on linear algebra and matrix theory. While it is self contained, it will work best for those who have already had some exposure to linear algebra. It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however.

This book features an ugly, elementary, and complete treatment of determinants early in the book. Thus it might be considered as Linear algebra done wrong. I have done this because of the usefulness of determinants. However, all major topics are also presented in an alternative manner which is independent of determinants.

The book has an introduction to various numerical methods used in linear algebra. This is done because of the interesting nature of these methods. The presentation here emphasizes the reasons why they work. It does not discuss many important numerical considerations necessary to use the methods effectively. These considerations are found in numerical analysis texts.

After being traditionally published for many years, this formidable text by W. Keith Nicholson is now being released as an open educational resource and part of Lyryx with Open Texts! Supporting today’s students and instructors requires much more than a textbook, which is why Dr. Nicholson opted to work with Lyryx Learning.

Overall, the aim of the text is to achieve a balance among computational skills, theory, and applications of linear algebra. It is a relatively advanced introduction to the ideas and techniques of linear algebra targeted for science and engineering students who need to understand not only how to use these methods but also gain insight into why they work.

The contents have enough flexibility to present a traditional introduction to the subject, or to allow for a more applied course. Chapters 1–4 contain a one-semester course for beginners whereas Chapters 5–9 contain a second semester course. The text is primarily about real linear algebra with complex numbers being mentioned when appropriate (reviewed in Appendix A).

Linear Transformations on Vector Spaces serves primarily as a textbook for undergraduate Linear Algebra courses. While standard Linear Algebra books begin by focusing on solving systems of linear equations and associated procedural skills, our book begins by developing a conceptual framework for the topic using the central objects, vector spaces and linear transformations. It covers the same concepts, skills, and, applications as conventional texts in a one-semester course, but students walk away with a much richer and more useful mastery of the topics. The book is structured to facilitate the implementation of the flipped classroom. The text features a continuous narrative to illuminate the big picture of the material and is written to help students develop their textbook reading skills. Also, there are “Explorations” scattered throughout each section; these are quick first examples intended for students to complete while reading before class meetings. Additional materials include section overview homework assignments and worksheets that can be used for in-class practice.