Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

Math 540: Linear Algebra II

Course description: This course is a natural continuation of Math 341. It covers topics in linear algebra which are essential for higher level courses in algebra and analysis. Topics include linear transformations, duality. Diagonalization of linear transformations.The Cayley Hamilton Theorem. Minimal polynomials. The Jordan canonical form. Exponential function. Inner product spaces, orthonormal bases and Gram-Schmidt orthogonalization. Operators on inner product spaces. Self-adjoint, unitary and positive operators. spectral theorem. Singular value decomposition. Bilinear and quadratic forms. Norms, bounded linear operators, matrix norms. Basic multilinear algebra.


Possible alternatives: Math 443 or Math 513 for a more applied treatment of some of the covered topics. 


Course content:


  • Review of linear transformations, duality.
  • Change of bases.

I. Diagonalization and related topics.

  •  Eigenvectors and eigenvalues.
  • Diagonalization of linear transformations and matrices.
  • Decompositions of polynomials, irreducibility, upper triangular matrices.
  • The Cayley-Hamilton Theorem.
  • Minimal polynomials.
  • The Jordan canonical form.
  • The exponential function.

II. Inner product spaces.

  • Inner products, Cauchy-Schwarz inequality, orthogonality.
  • Orthonormal sets and bases, projections, Gram-Schmidt orthogonalization.
  • Adjoints, self-adjoint transformations, spectral theorem, commuting transformations.
  • Operators on inner product spaces, orthogonal and unitary maps and matrices. normal transforma- tions, spectral theorem. Positive operators and polar decomposition.
  • Singular value decomposition.

III. Bilinear and Quadratic forms

  •  Matrix representation of bilinear forms. Symmetric bilinear forms.
  •  Signature, Sylvester’s theorem, positive definite forms.
  •  Quadratic forms.

IV. Further topics

These are chosen by the instructor. Possible choices are

  • Bounded linear operators, norms, matrix norms.
  • Multilinear algebra: tensors, alternating forms.
  • Applications.
3 (N-A)
(MATH 341 or 375) or (MATH 320 and MATH 421, 521, 541, or 551) or (MATH 340 and MATH 421, 521, 541, or 551)

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, WI  53706

(608) 263-3054

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