Linear Algebra II

Spectral Theory and Abstract Vector Spaces

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154 pages
This contains the basic abstract theory of Linear algebra.
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Kenneth Kuttler received his Ph.D. in mathematics from The University of Texas at Austin in 1981. From there, he went to Michigan Tech. University where he was employed for most of the next 17 years. He joined the faculty of Brigham Young University in 1998 and has been there since this time. Kuttler'

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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  1. Vector Spaces And Fields
    1. Vector Space Axioms
    2. Subspaces And Bases
    3. Lots Of Fields
    4. Exercises
  2. Linear Transformations
    1. Matrix Multiplication As A Linear Transformation
    2. L (V, W) As A Vector Space
    3. The Matrix Of A Linear Transformation
    4. Eigenvalues And Eigenvectors Of Linear Transformations
    5. Exercises
  3. Canonical Forms
    1. A Theorem Of Sylvester, Direct Sums
    2. Direct Sums, Block Diagonal Matrices
    3. Cyclic Sets
    4. Nilpotent Transformations
    5. The Jordan Canonical Form
    6. Exercises
    7. The Rational Canonical Form
    8. Uniqueness
    9. Exercises
  4. Markov Processes
    1. Regular Markov Matrices
    2. Migration Matrices
    3. Absorbing States
    4. Exercises
  5. Inner Product Spaces
    1. General Theory
    2. The Gram Schmidt Process
    3. Riesz Representation Theorem
    4. The Tensor Product Of Two Vectors
    5. Least Squares
    6. Fredholm Alternative Again
    7. Exercises
    8. The Determinant And Volume
    9. Exercises
  6. Self Adjoint Operators
    1. Simultaneous Diagonalization
    2. Schur’s Theorem
    3. Spectral Theory Of Self Adjoint Operators
    4. Positive And Negative Linear Transformations
    5. The Square Root
    6. Fractional Powers
    7. Square Roots And Polar Decompositions
    8. An Application To Statistics
    9. The Singular Value Decomposition
    10. Approximation In The Frobenius Norm
    11. Least Squares And Singular Value Decomposition
    12. The Moore Penrose Inverse
    13. Exercises
  • Index