Essential Linear Algebra
by Jared M. Maruskin
264 pages.
published December 3, 2012.
ISBN: 978-0-9850627-3-6 (hardcover)
About the Book
This text introduces linear algebra—boiled to its essence—presented in a clear and concise fashion. Designed around a single-semester undergraduate course, Essential Linear Algebra introduces key concepts, various real-world applications, and provides detailed yet understandable proofs of key results that are aimed towards students with no advanced preparation in proof writing. The level of sophistication gradually increases from beginning to end in order to prepare students for subsequent studies.
We begin with a detailed introduction to systems of linear equations and elementary row operations. We then advance to a discussion of linear transformations, which provide a second, more geometric, interpretation of the operation of matrix-vector product. We go on to introduce vector spaces and their subspaces, the image and kernel of a transformation, and change of coordinates. Following, we discuss matrices of orthogonal projections and orthogonal matrices. Our penultimate chapter is devoted to the theory of determinants, which are presented, first, in terms of area and volume expansion factors of 2x2 and 3x3 matrices, respectively. We use a geometric understanding of volume in n-dimensions to introduce general determinants axiomatically as multilinear, antisymmetric mappings, and prove existence and uniqueness. Our final chapter is devoted to the theory of eigenvalues and eigenvectors. We conclude with a number of discussions on various types of diagonalization: real, complex, and orthogonal.
Table of Contents (Back to Top)
Chapter 1 Linear Systems
1.1 Linear Geometry
1.2 Modeling and Linear Systems
1.3 Matrix Representation
1.4 Gauss--Jordan Elimination
1.5 Solution Sets for Linear Systems
Chapter 2 Linear Transformations
2.1 Linear Transformations and Matrices
2.2 Matrix Products
2.3 Inverse Matrices
Chapter 3 Vector Spaces
3.1 Vector Spaces and Subspaces
3.2 Linear Independence, Basis, and Dimension
3.3 Image and Kernel of a Linear Transformation
3.4 Coordinates
3.5 Matrix Transpose: Row Space and Column Space
Chapter 4 Orthogonality
4.1 Orthogonal Projections
4.2 The Gram--Schmidt Process
4.3 Orthogonal Transformations and Their Matrices
4.4 Least Squares
Chapter 5 Determinants
5.1 Determinants in Two and Three Dimensions
5.2 Determinants Defined
5.3 The Gram Determinant
Chapter 6 Eigenvalues and Eigenvectors
6.1 Eigenvalues
6.2 Eigenvectors
6.3 Diagonalization
6.4 Complex Eigenvalues and Eigenvectors
6.5 Symmetric Matrices
