Advanced linear algebra /
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York :
Springer-Verlag,
©1992.
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| Series: | Graduate texts in mathematics ;
135. |
| Subjects: |
Table of Contents:
- Ch. 0. Preliminaries. pt. 1. Preliminaries. Matrices. Determinants. Polynomials. Functions. Equivalence Relations. Zorn's Lemma. Cardinality. pt. 2. Algebraic Structures. Groups. Rings. Integral Domains. Ideals and Principal Ideal Domains. Prime Elements. Fields. The Characteristic of a Ring
- pt. 1. Basic Linear Algebra. Ch. 1. Vector Spaces. Vector Spaces. Subspaces. The Lattice of Subspaces. Direct Sums. Spanning Sets and Linear Independence. The Dimension of a Vector Space. The Row and Column Space of a Matrix. Coordinate Matrices. Exercises. Ch. 2. Linear Transformations. Linear Transformations. The Kernel and Image of a Linear Transformation. Isomorphisms. The Rank Plus Nullity Theorem. Linear Transformations from F[superscript n] to F[superscript m]. Change of Basis Matrices. The Matrix of a Linear Transformation. Change of Bases for Linear Transformations. Equivalence of Matrices. Similarity of Matrices. Invariant Subspaces and Reducing Pairs. Exercises. Ch. 3. The Isomorphism Theorems. Quotient Spaces. The First Isomorphism Theorem. The Dimension of a Quotient Space. Additional Isomorphism Theorems. Linear Functionals. Dual Bases. Reflexivity. Annihilators. Operator Adjoints. Exercises.
- Ch. 9. Real and Complex Inner Product Spaces. Introduction. Norm and Distance. Isometries. Orthogonality. Orthogonal and Orthonormal Sets. The Projection Theorem. The Gram-Schmidt Orthogonalization Process. The Riesz Representation Theorem. Exercises. Ch. 10. The Spectral Theorem for Normal Operators. The Adjoint of a Linear Operator. Orthogonal Diagonalizability. Motivation. Self-Adjoint Operators. Unitary Operators. Normal Operators. Orthogonal Diagonalization. Orthogonal Projections. Orthogonal Resolutions of the Identity. The Spectral Theorem. Functional Calculus. Positive Operators. The Polar Decomposition of an Operator. Exercises
- pt. 2. Topics. Ch. 11. Metric Vector Spaces. Symmetric, Skew-symmetric and Alternate Forms. The Matrix of a Bilinear Form. Quadratic Forms. Linear Functionals. Orthogonality. Orthogonal Complements. Orthogonal Direct Sums. Quotient Spaces. Symplectic Geometry
- Hyperbolic Planes. Orthogonal Geometry
- Orthogonal Bases. The Structure of an Orthogonal Geometry. Isometries. Symmetries. Witt's Cancellation Theorem. Witt's Extension Theorem. Maximum Hyperbolic Subspaces. Exercises.