Math 700 — Linear Algebra — Spring 2014

Instructor: Dr. William DeMeo

Time/Place: MW 3:55–5:10, LeConte Rm 310

Textbook: Linear Algebra and Its Applications, 2nd Edition, Peter Lax.

Other References:

  1. Finite Dimensional Vector Spaces, Paul Halmos.
  2. Linear Algebra Problem Book, Paul Halmos.
  3. Advanced Linear Algebra, Steven Roman.

Brief Summary: (from the catalog) Vector spaces, linear transformations, dual spaces, decompositions of spaces, and canonical forms.

Details: This is a beginning graduate level treatment of linear algebra in which we focus on finite dimensional vector spaces from a more general point of view than you might have seen in an undergraduate class. Below is a detailed list of some of the topics we will cover (essentially the first 172 pages of Lax’s book, plus additional topics depending on students’ interests). In addition to these topics, we will cover vector spaces over finite fields, the Jordan canonical form and Smith normal form. Depending on student interest, we might also spend some time demonstrating and applying the theory using computer algebra software (like Sage) and numerical software (like Matlab or Octave/Sage).

  1. Review of Fundamentals: Linear Space, Isomorphism; Subspace; Linear Dependence; Basis, Dimension; Quotient Space.
  2. Duality: Linear Functions; Dual of a Linear Space; Annihilator; Codimension; Quadrature Formula.
  3. Linear Mappings: Domain and Target Space; Nullspace and Range; Fundamental Theorem; Underdetermined Linear Systems; Interpolation; Difference Equations; Algebra of Linear Mappings; Dimension of Nullspace and Range; Transposition; Similarity; Projections.
  4. Matrices: Rows and Columns, Matrix Multiplication, Transposition, Rank, Gaussian Elimination
  5. Determinant and Trace: Ordered Simplices; Signed Volume, Determinant; Permutation Group; Formula for Determinant; Multiplicative Property; Laplace Expansion; Cramer’s Rule; Trace
  6. Spectral Theory: Iteration of Linear Maps; Eigenvalues, Eigenvectors; Fibonacci Sequence; Characteristic Polynomial; Trace and Determinant Revisited; Spectral Mapping Theorem; Cayley-Hamilton Theorem; Generalized Eigenvectors; Spectral Theorem; Minimal Polynomial; When Are Two Matrices Similar; Commuting Maps.
  7. Euclidean Structure: Scalar Product, Distance; Schwarz Inequality; Orthonormal Basis; Gram-Schmidt; Orthogonal Complement; Orthogonal Projection; Adjoint; Overdetermined Systems; Isometry; The Orthogonal Group; Norm of a Linear Map; Completeness Local Compactness; Complex Euclidean Structure; Spectral Radius; Hilbert-Schmidt Norm; Cross Product.
  8. Spectral Theory of Self-Adjoint Mappings: Quadratic Forms; Law of Inertia; Spectral Resolution; Commuting Maps; Anti-Self-Adjoint Maps; Normal Maps; Rayleigh Quotient; Minmax Principle; Norm and Eigenvalues.
  9. Calculus of Vector- and Matrix-Valued Functions: Convergence in Norm; Rules of Differentiation, Derivative of det A(t), Matrix Exponential; Simple Eigenvalues; Multiple Eigenvalues; Rellich’s Theorem; Avoidance of Crossing.
  10. Matrix Inequalities: Positive Self-Adjoint Matrices; Monotone Matrix Functions; Gram Matrices; Schur’s Theorem; The Determinant of Positive Matrices; Integral Formula for Determinants; Eigenvalues; Separation of Eigenvalues; Wielandt-Hoffman Theorem; Smallest and Largest Eigenvalue; Matrices with Positive Self-Adjoint Part; Polar Decomposition; Singular Values; Singular Value Decomposition;

And a selection of topic from the following, as time permits and depending on students’ interests:

  • Convexity: Convex Sets; Gauge Function; Hahn-Banach Theorem; Support Function; Caratheodory’s Theorem; Konig-Birkhoff Theorem; Helly’s Theorem.
  • The Duality Theorem: Farkas-Minkowski Theorem; Duality Theorem; Economics Interpretation; Minmax Theorem.
  • Normed Linear Spaces: Norm; lp Norms; Equivalence of Norms; Completeness; Local Compactness; Theorem of F. Riesz; Dual Norm; Distance from Subspace; Normed Quotient Space; Complex Normed Spaces; Complex Hahn-Banach Theorem; Characterization of Euclidean Spaces.
  • Linear Mappings Between Normed Linear Spaces: Norm of a Mapping; Norm of Transpose; Normed Algebra of Maps; Invertible Maps; Spectral Radius;
  • Positive Matrices: Perron’s Theorem; Stochastic Matrices; Frobenius’ Theorem.
  • How to Calculate the Eigenvalues of Self-Adjoint Matrices: QR Factorization; Using the QR Factorization to Solve Systems of Equations; The QR Algorithm for Finding Eigenvalues; Householder Reflection for OR Factorization; Tridiagonal Form; Analogy of QR Algorithm and Toda Flow; Moser’s Theorem.
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