Because this material is spread over several weeks, students have more time than usual to absorb and review these critical concepts. Orthogonality and Least-Squares Problems — These topics receive a more comprehensive treatment than is commonly found in beginning texts. The Fifth Edition linear algebra includes additional support for concept and proof based learning. Conceptual Practice Problems and their solutions have been added so that most sections now have a proofs or concept based examples for students to review.
Additional guidance has also been added to some of the proofs of theorems in the body of this Linear Algebra textbook. As in the first three editions, I have taken into account the reality that students taking introductory linear algebra are likely to come from a variety of disciplines.
In addition to mathematics majors, there are apt to be majors from engineering, physics, chemistry, computer science, biology, environmental science, geography, economics, psychology, business, and education, as well as other students taking the course as an elective or to fulfill degree requirements. Accordingly, the book balances theory and applications, is written in a conversational style yet is fully rigorous, and combines a traditional presentation with concern for student-centered learning.
There is no such thing as a universally best learning style. In any class, there will be some students who work well independently and others who work best in groups; some who prefer lecture-based learning and others who thrive in a workshop setting, doing explorations; some who enjoy algebraic manipulations, some who are adept at numerical calculations with and without a computer , and some who exhibit strong geometric intuition.
In this edition, I continue to present material in a variety of ways-algebraically, geometrically, numerically, and verbally-so that all types oflearners can find a path to follow. I have also attempted to present the theoretical, computational, and applied topics in a flexible yet integrated way.
In doing so, it is my hope that all students will be exposed to the many sides of linear algebra. From a pedagogical point of view, there is no doubt that for most students believe strongly that linear algebra is essentially about vectors and that students need to see vectors first in a concrete setting in order to gain some geometric insight. Moreover, introducing vectors early allows students to see how systems of linear equations arise naturally from geometric problems.
Matrices then arise equally naturally as coefficient matrices oflinear systems and as agents of change linear transformations. This sets the stage for eigenvectors and orthogonal projections, both of which are best understood geometrically.
The dart that appears on the cover of this book symbolizes a vector and reflects my conviction that geometric understanding should precede computational techniques.
The gears change in Chapter 2 as students reach the introduction of vector spaces. Throughout the book, the theory is motivated and reinforced by genuine applications, allowing pure mathematicians to teach applied mathematics.
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