Synopses & Reviews
For courses in Advanced Linear Algebra. This top-selling, theorem-proof text presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. - NEW - Added section on the singular value decomposition. - Discusses the pseudoinverse of a matrix or a linear transformation between finite-dimensional inner product spaces. - NEW - Revised proofs, added examples and exercises. - Improves the clarity of the text and enhances students understanding of it. - Numerous accessible exercises - Enriches and extends the text material. - Offers students a chance to test their understanding by working interesting problems at a reasonable level of difficulty. - Real-world applications throughout. - Reveals to students the power of the subject by demonstrating its practical uses.
Synopsis
This top-selling, theorem-proof book presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Chapter topics cover vector spaces, linear transformations and matrices, elementary matrix operations and systems of linear equations, determinants, diagonalization, inner product spaces, and canonical forms. For statisticians and engineers.
Table of Contents
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 Equations—Theoretical Aspects. Systems of Linear Equations—Computational Aspects.
4. Determinants.
Determinants of Order 2. Determinants of Order n. Properties of Determinants. Summary—Important Facts about Determinants. A Characterization of the Determinant.
5. Diagonalization.
Eigenvalues and Eigenvectors. Diagonalizability. Matrix Limits and Markov Chains. Invariant Subspaces and the Cayley-Hamilton Theorem.
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. The Singular Value Decomposition and the Pseudoinverse. Bilinear and Quadratic Forms. Einstein's Special Theory of Relativity. Conditioning and the Rayleigh Quotient. The Geometry of Orthogonal Operators.
7. Canonical Forms.
The Jordan Canonical Form I. The Jordan Canonical Form II. The Minimal Polynomial. Rational Canonical Form.
Appendices.
Sets. Functions. Fields. Complex Numbers. Polynomials.
Answers to Selected Exercises.
Index.