Midterm ExamΒΆ

The in-class Midterm exam will be on Wednesday November 16, 2:30 - 3:20pm.

The exam is closed book, closed notes. No calculators.

The exam will cover roughly Lectures 1-15 in Trefethen and Bau.

Note:

  • There is no lecture on Friday, November 11 (Veterans’ Day)
  • There will be no in-class lecture on Monday, November 14. Instead a video lecture will be posted that will primarily review some of the key concepts. See the Canvas page
  • Some extra office hours will be scheduled, but don’t put off understanding this material to the last minute.

Office hours

LeVeque will have office hours

  • in Lewis 328

    • Thursday 11/10, 2:30 - 3:30pm
    • Tuesday 11/15, 5:00 - 6:00pm

  • On GoToMeeting

    • Sunday 11/13, 12:00 - 2:00pm PST
    • Tuesday 11/15, 7:00 - 8:00am PST

Some key concepts and algorithms that you should know:

  • Basic linear algebra concepts we have used, e.g. linear spaces, basis for a space, inner and outer products, orthogonality, vector and matrix norms.
  • Formulas for computing 1-norm and \(\infty\)-norm of vectors and matrices. Formula for 2-norm of vector, Frobenius norm of matrix, and 2-norm of matrix (in terms of singular values).
  • The form of the SVD for a general \(m\times n\) matrix (reduced form and full). What properties the matrices \(U, \Sigma, V\) have.
  • Finding an SVD by hand of very simple matrices, as in some of the homework problems.
  • Form of the QR factorization of a general matrix (reduced form and full), and what properties these factors have.
  • Classical Gram-Schmidt orthogonalization of a set of vectors (e.g. columns of a matrix) and how this is used to compute the QR factorization of a full rank matrix.
  • You don’t have to memorize the modified Gram-Schmidt algorithm, but you should understand it.
  • Properties of projection matrices and orthogonal projections.
  • Determining the projection matrix for orthogonal projection onto the one-dimensional subspace described as the span of a given nonzero vector.
  • Determining the projection matrix for orthogonal projection onto the range of a matrix with full column rank, using the SVD or the QR factorization.
  • You should understand reflectors and the ideas of the Householder triangularization algorithm for computing a QR factorization, but you don’t have to memorize the formulas.
  • Least squares problems: The normal equations, and how to compute the solution using the QR factorization or the SVD of a matrix with full column rank.
  • Setting up a least squares problem, e.g. “find the function of a given form (in terms of some specified basis functions) that gives the best fit to some set of data in the least squares sense”. (Setting up the matrix and right hand side).
  • The condition number of a problem vs. stability of an algorithm – basic ideas.
  • The condition number of a matrix, \(\kappa(A)\), how to compute it (in terms of norms, or in terms of singular values for the 2-norm case), what it means in terms of bounds on the relative errors in a problem solution when the data is perturbed.