Spectral Theory and Abstract Vector Spaces

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208 pages

Language:

English

This contains the basic abstract theory of Linear algebra.

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Description

Content

This contains the basic abstract theory of Linear algebra. It includes a discussion of general fields of scalars, spectral theory, canonical forms, applications to Markov processes, and inner product spaces.

- Spectral Theory
- Eigenvalues
- Some Applications Of Eigenvalues And Eigenvectors
- Exercises
- Schurâ€™s Theorem
- Trace And Determinant
- Quadratic Forms
- Second Derivative Test
- The Estimation Of Eigenvalues
- Advanced Theorems
- Exercises

- Vector Spaces And Fields
- Vector Space Axioms
- Subspaces And Bases
- Lots Of Fields
- Exercises

- Linear Transformations
- Matrix Multiplication As A Linear Transformation
- L(V,W) As A Vector Space
- The Matrix Of A Linear Transformation
- Eigenvalues And Eigenvectors Of Linear Transformations
- Exercises

- Linear Transformations Canonical Forms
- A Theorem Of Sylvester, Direct Sums
- Direct Sums, Block Diagonal Matrices
- Cyclic Sets
- Nilpotent Transformations
- The Jordan Canonical Form
- Exercises
- The Rational Canonical Form
- Uniqueness
- Exercises

- Markov Chains And Migration Processes
- Regular Markov Matrices
- Migration Matrices
- Markov Chains
- Exercises

- Inner Product Spaces
- General Theory 181
- The Gram Schmidt Process
- Riesz Representation Theorem
- The Tensor Product Of Two Vectors
- Least Squares
- Fredholm Alternative Again
- Exercises
- The Determinant And Volume
- Exercises