The review will include: introduction of the real numbers through Dedekind cuts, continuity of real-valued functions on the real line Cantor nested-interval principle, basic results for continuous functions, Maximum and Intermediate Value theorems, Heine-Borel Theorem, Uniform Continuity on closed intervals metric spaces, convergence of sequences, Cauchy sequences, completeness, more general uniform continuity and intermediate value theorems general topology, separation, compactness, product spaces, Tychonoff's Theorem. Mathematics 4800 will open with a review of the basics of real analysis (brief or extended background requires). The ideas and tools provided by this course will be useful to students who intend to tackle higher level courses in digital signal processing, computer vision, robotics, and computer graphics.
Applications will include the use of matrix computations to computer graphics, use of the discrete Fourier transform and related techniques in digital signal processing, the analysis of systems of linear differential equations, and singular value deompositions with application to a principal component analysis. Some specific topics: the solution of systems of linear equations by Gaussian elimination, dimension of a linear space, inner product, cross product, change of basis, affine and rigid motions, eigenvalues and eigenvectors, diagonalization of both symmetric and non-symmetric matrices, quadratic polynomials, and least squares optimazation. This course will introduce students to some of the most widely used algorithms and illustrate how they are actually used. Many important problems in a wide range of disciplines within computer science and throughout science are solved using techniques from linear algebra.