Introduction to Linear and Matrix Algebra

Introduction to Linear and Matrix Algebra coverThis textbook emphasizes the interplay between algebra and geometry to motivate the study of linear algebra. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. By focusing on this interface, the author offers a conceptual appreciation of the mathematics that is at the heart of further theory and applications. Those continuing to a second course in linear algebra will appreciate the companion volume Advanced Linear and Matrix Algebra.

Starting with an introduction to vectors, matrices, and linear transformations, the book focuses on building a geometric intuition of what these tools represent. Linear systems offer a powerful application of the ideas seen so far, and lead onto the introduction of subspaces, linear independence, bases, and rank. Investigation then focuses on the algebraic properties of matrices that illuminate the geometry of the linear transformations that they represent. Determinants, eigenvalues, and eigenvectors all benefit from this geometric viewpoint. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from linear programming, to power iteration and linear recurrence relations. Exercises of all levels accompany each section, including many designed to be tackled using computer software.

Introduction to Linear and Matrix Algebra is ideal for an introductory proof-based linear algebra course. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. Students are assumed to have completed one or two university-level mathematics courses, though calculus is not an explicit requirement. Instructors will appreciate the ample opportunities to choose topics that align with the needs of each classroom, and the online homework sets that are available through WeBWorK.

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  • From zbMATH (review by Qing-Wen Wang):

    This book is easy to read and requires almost no basic prior knowledge. It presents a preliminary course in linear algebra, along with an introduction to the interplay between algebra and geometry. Quite different from traditional textbooks, it begins with vectors, matrix algebra and linear transformations before moving into the advanced topics. It also includes many computations, both as examples and as exercises (some of those are designed to let readers see how computer programming can be used as a problem-solving tool for algebra exercises).

    . . .

    This is a good textbook on linear and matrix algebra. I recommend this book to the people who is learning linear algebra for the first time.

  • From the paper “Various Course Proposals for: Mathematics with a View Towards (the Theoretical Underpinnings of) Machine Learning” by Marc S. Paolella:

    The optical presentation is among the best I have ever seen in a textbook, with the author having used some color and judicious spacing. The layout (font, everything) is beautiful. . . . [I] can say that, pedagogically, [Johnston’s books] seem outstanding.

  • From the Mathematical Gazette (review by Owen Toller, page 1, page 2):

    The writing is accessible, indeed very readable . . ., but it does not lack rigour.

    . . .

    It seems to me an excellent first course for undergraduates without a strong mathematical background who are taking subjects where the technicalities of linear algebra are needed.

    . . .

    Above all, any schools where further mathematics is taught would be well advised to have a copy for departmental use; teachers who read it would feel more confident in their teaching, and I am sure they would enjoy the experience.

Citation/bibliographic information:

  • Author: Nathaniel Johnston
  • Cite as: N. Johnston. Introduction to Linear and Matrix Algebra. Springer International Publishing, 2021.
  • DOI: 10.1007/978-3-030-52811-9
  • eBook ISBN: 978-3-030-52811-9
  • Hardcover ISBN: 978-3-030-52810-2
  • BibTeX entry: 
    @book{johnston2021introduction,
      title={Introduction to Linear and Matrix Algebra},
      author={Nathaniel Johnston},
      isbn={978-3-030-52810-2},
      url={https://www.springer.com/us/book/9783030528102},
      year={2021},
      publisher={Springer International Publishing}
}