Rating:

( 15 )

154 pages

Language:

en

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.

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- 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

- 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 Processes
- Regular Markov Matrices
- Migration Matrices
- Absorbing States
- Exercises

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

- Self Adjoint Operators
- Simultaneous Diagonalization
- Schur’s Theorem
- Spectral Theory Of Self Adjoint Operators
- Positive And Negative Linear Transformations
- The Square Root
- Fractional Powers
- Square Roots And Polar Decompositions
- An Application To Statistics
- The Singular Value Decomposition
- Approximation In The Frobenius Norm
- Least Squares And Singular Value Decomposition
- The Moore Penrose Inverse
- Exercises

- Index