Preface Preface and Notes to Students and Instructors
Linear Algebra is the study of vectors, objects that can be 1) scaled by multiplication and 2) added, and functions which respect those operations called linear functions which lead to linear equations. In this text we emphazise the algebra in linear algebra. For any given type of vector (and there are many) and a given linear equation, the solution set is always a “flat” object: a line, a plane, or a higher-dimensional object called a hyperplane, in the appropriate space. We will not emphasize the geometry of these objects, but we will appeal to it from time to time to help develop our intuition. Our aim is
Except for material in Chapter 5 we will restrict ourselves to vectors constructed as ordered lists of real numbers. So this course covers real linear algebra, with no complex numbers and minimal abstraction. Though the course is largely computational, there are numerous opportunities for students to engage in proofs, both within and outside of Chapter 5.
The chapters are: Vectors and Matrices, Elimination and Applications, The Determinant, Vector Spaces, Matrix-Based Vector Spaces, Projections, Eigenvalues and Eigenvectors, and Linear Transformations. For each topic within a chapter, the text has three sections: content, paired with a worksheet and a homework set. Throughout the content sections, examples are offered. These examples, as well as proofs to many theorems, are initially hidden but may be revealed by single-clicking.
The worksheets are meant to 1) provide students a chance to work in class on developing concept images and familiarity with any algorithmic processes before starting on the homework, and 2) provide the instructor with an opportunity to offer formative assessment.
Every concept in this text is specified as a numbered definition, theorem, example, algorithm or remark.
We have made an effort to ensure sensible flow of the material, and have placed significant asides in the Appendix.
When we prove theorems or make logical arguments using theorems we will always cite the relevant theorems, but when we compute we will decline to cite theorems which tell us how to compute.
Notation is critical; it should be unambiguous, consistent, sensible / organic, and not impede learning. The reader will likely find a small number of new notations, which we have introduced after very careful consideration and which are consistent, organic, and will only enhance learning.
- \(\displaystyle \R^n,\,\R^{m\times n}\)
- Matrices are in bold, vectors are in bold and have an arrow above (except for vectors which are rows or columns of matrices), scalars are not bold, \(\A,\,\A^{i\ast},\,\A_{\ast j},\,\x,\,A_{ij}\)
- \(\displaystyle \E_{(ij)},\,\E^{(a)ii},\,\E_{(a)ji}\)
- Parallelity, antiparallelity, \(\parallel,\,\upharpoonleft\!\downharpoonright\)
Students: Diving in to the worksheet problems will improve your level of preparation prior to working the associated homework problems. You may find that the algebra itself is not too taxing, but there is plenty of it. You will see a bit of trigonometry but little calculus.
Instructors: This is a sophomore-level course, loosely speaking. Mathematically mature students who have taken trigonometry but not yet taken calculus may be welcomed. At WOU our courses are 10 weeks; we are typically able to cover Chapter 2, Chapter 3 except for the LU-decomposition, Chapter 4, Chapter 5, Section 6.1 in Chapter 6, Chapter 8, and Chapter 9. With the inclusion of the chapters and sections omitted from the list above, the text is appropriate for a semester-length course.
For ten-week courses the following chapters / sections may be omitted without loss of continuity: the LU-Decomposition, Constructing Matrices with Given Row, Column, or Nullspace Properties, Similarity, Projections, and Linear Transformations.
The nature of the material permits the instructor to choose the role proofs may play in the course content. The text works equally well whether your course emphasizes computation, proof or both.
The worksheets may be completed in groups or individually, and were designed with the assumption that while engaging the worksheets students will have access to this text. While teaching this course at WOU, we split class time roughly evenly between lecture and in-class work. By that standard most worksheets contain an excess of problems so instructors may customize their approach. This text is the product of years of work but still there will be typos and improvements to be made; if you find any we would be grateful if you would let us know.
All: We have chased down a lot of typos but rest assured some remain. Should you find any, we would be grateful if you would email us at at beavers@wou.edu or beaverc@wou.edu and we will correct them within a few days.
November 29, 202314:09:05 (-08:00)
