This sheet is available at www.math.ust.hk/~amoy/math217

Math 217 - Fall 2010


Tentative syllabus of topics.

Week one: Set and functions; Groups and homomorphisms : (Sep 01)
  • Set and functions
  • Composition of functions
  • Inverse function, permutations

    Week two: Groups, homomorphisms, fields, rings: (Sep 06)
  • Groups, subgroups, homomorphisms, kernel, image
  • Fields: real, complex, rational, finite
  • Rings

    Week three: Vector spaces, subspaces, and linear transformations: (Sep 13)
  • Vector spaces, span, subspaces
  • Linear transformations
  • Kernel and image of a linear map

    Week four: (Sep 20)
  • Composition of linear maps
  • The ring of linear transformations of a vector space to itself
  • Direct Sums
  • Linear independence, basis

    Week five: Dimension: (Sep 27)
  • Basis, dimension
  • Sets of linear transformations as vector spaces
  • Rank/Nullity Theorem
  • Coordinates

    Week six: Linear Mappings and matrix algebra: (Oct 04)
  • Isomorphisms and bases
  • Matrix arithmetic
  • Representation of linear map by a matrix

    Week seven. Computational algorithms: (Oct 11)
  • Change of coordinate matrix
  • Linear systems and matrices
  • Row operations and Gaussian elimination
  • Maxima software

    Week eight. Dot products in R^n: (Oct 18)
  • Dot products in R^n
  • The matrix of an orthogonal projection
  • Proof that the matrix of a projection is symmetric

    Week nine. Inner product spaces: (Oct 25)
  • Real inner product spaces
  • Change of coordinates matrices for two orthonormal bases
  • Orthogonal bases and projections
  • Complex inner product spaces

    Week ten. Determinants: (Nov 01)
  • The vector space of multilinear maps
  • Existence of determinant and basic properties
  • Characteristic polynomial

    Week eleven. Eigenvalues and eigenvectors: (Nov 08)
  • Eigenvalues and eigenvectors
  • Computations

    Week twelve. Spectral Theory: (Nov 15)
  • Self-adjoint linear maps
  • Spectral theorem for self adjoint maps, complex case
  • Spectral theorem for self adjoint maps, real case
  • Spectral theorem for unitary transformations
  • Spectral theorem for normal transformations

    Week thirteen. Cayley-Hamilton (Nov 22)
  • Invariant subspaces
  • Cayley-Hamilton theorem

    Week fourteen. Jordan Decomposition: (Nov 29)
  • Ideals of the rings Z and K[x]
  • Jordan Decomposition