Document Type

Working Paper

Publication Date

7-23-2015

Abstract

Table of Contents:

1. Systems of Linear Equations and Matrices

  • Systems of Linear Equations
  • Elementary Row Operations
  • Row Reduction and Reduced Row-Echelon Form
  • Solutions of Systems of Linear Equations
  • Matrix Operations
  • Matrix Inverses

2. Euclidean 2-Space and 3-Space

  • Vectors in the Plane and in Space
  • The Dot Product
  • Cross Product
  • Lines in Space
  • Planes in Space

3. Determinants

  • The Definition of Determinant
  • Elementary Row Operations and the Determinant
  • Elementary Matrices and the Determinant
  • Applications of the Determinant

4. Vector Spaces and Subspaces

  • Vector Spaces
  • Subspaces
  • Linear Dependence and Independence
  • Basis and Dimension

5. Linear Transformations

  • Definition of Linear Transformation
  • The Kernel and Range of a Linear Transformation
  • Isomorphisms and Coordinate Vectors
  • Similarity

6. Eigenvalues and Eigenvectors

  • Eigenvalues, Eigenvectors, and Eigenspaces
  • The Characteristic Polynomial
  • Diagonalization
  • Eigenvalues and Linear Operators

7. (Real) Inner Product Spaces

  • Introduction
  • Angle and Orthogonality
  • Orthogonal and Orthonormal Bases
  • Gram-Schmidt
  • Least Squares
  • Inner Product Space Isomorphisms and Orthogonal Matrices
  • Symmetric Matrices and Orthogonal Diagonalizability

8. Applications

  • Quadratic Forms
  • Constrained Optimization of Quadratic Forms
  • Conic Sections
  • Quadric Surfaces
  • Positive Definite Matrices
  • Singluar Value Decomposition
  • Matrix Norms
  • The Pseudoinverse

Selected Answers

Index

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