An accessible, applications-oriented presentation of the theory of
linear algebra. This is the top selling theorem-proof text in the
A significant number of interesting and accessible exercises.
Real-world applications are provided throughout the book
that reveal the power of the subject by demonstrating its practical
NEWThe text has been extensively rewritten with
many additional examples and exercises and for increased clarity.
NEWAdditional material on the reduced echelon
form of a matrix is included in Section 3.4.
1. Vector Spaces.
Introduction. Vector Spaces. Subspaces. Linear Combinations
and Systems of Linear Equations. Linear Dependence and Linear Independence.
Bases and Dimension. Maximal Linearly Independent Subsets.
2. Linear Transformations and Matrices.
Linear Transformations, Null Spaces, and Ranges. The Matrix
Representation of a Linear Transformation. Composition of Linear Transformations
and Matrix Multiplication. Invertibility and Isomorphisms. The Change
of Coordinate Matrix. Dual Spaces. Homogeneous Linear Differential
Equations with Constant Coefficients.
3. Elementary Matrix Operations and Systems of Linear Equations.
Elementary Matrix Operations and Elementary Matrices. The
Rank of a Matrix and Matrix Inverses. Systems of Linear EquationsTheoretical
Aspects. Systems of Linear EquationsComputational Aspects.
Determinants of Order 2. Determinants of Order n.
Properties of Determinants. SummaryImportant Facts about Determinants.
A Characterization of the Determinant.
Eigenvalues and Eigenvectors. Diagonalizability. Matrix
Limits and Markov Chains. Invariant Subspaces and the Cayley-Hamilton
6. Inner Product Spaces.
Inner Products and Norms. The Gram-Schmidt Orthogonalization
Process and Orthogonal Complements. The Adjoint of a Linear Operator.
Normal and Self-Adjoint Operators. Unitary and Orthogonal Operators
and Their Matrices. Orthogonal Projections and the Spectral Theorem.
Bilinear and Quadratic Forms. Einstein's Special Theory of Relativity.
Conditioning and the Rayleigh Quotient. The Geometry of Orthogonal
7. Canonical Forms.
Jordan Canonical Form I. Jordan Canonical Form II. The Minimal
Polynomial. Rational Canonical Form.