Bootcamp 2011 - Linear Algebra


The Linear Algebra part of the bootcamp meets M-Th 2-4pm in MS6627. My office hours are M-Th 1-2pm in MS7901.

Homework Assignments


Homework is due in the TA section every Friday. The TA will hand you back the graded homework during the subsequent week. Some homework problems are taken from old basic exams. These problems are referenced as XNNMM, where X is one of F, W, S (meaning Fall, Winter or Spring, respectively), NN is a two-digit year of the exam, and MM is the number of the problem on that exam. You may download copies of past basic exams on the web from the following link: http://www.math.ucla.edu/grad/handbook/hbquals.shtml
Problems marked with an asterisk (*) are optional.

  1. S0208; S0209; W0209 (recall that v is called an eigenvector of a linear transformation T with eigenvalue a if v is nonzero and Tv = av); F0108; S0307; S0501*; S0502(a); F0509*; F0610(b)*; S0703
  2. F0107; W0208; S0308; F0310; F0409; F0410; F0510; S0702*; S1101; S1102; F1005; F1006
  3. F0110; F0209; F0210; W0210; W0211; S0309; S0310; F0309*; S0410*; S049; F0611; S1001; S1002. Also: Problem A. Let T be a linear transformation from V to W so that all eigenvalues of T*T lie between 0 and 1. Show that there exists a vector space U, an isometry A from V to U and a co-isometry B from U to W so that T=AB. Conversely, show that if T=AB with A isometry and B coisometry, then all eigenvalues of T*T lie between 0 and 1. Problem B. For a positive-definite Hermitian matrix A let f(A) = Tr( log (A)) = sum of the logarithms of eigenvalues of A . Show that: (a) The set of all positive-definite Hermitian matrices is convex (i.e., if A and B are positive definite Hermitian, then so is t A+(1-t)B for all t between 0 and 1); (b) Show that f is a strictly concave function (that is, if C= t A+(1-t)B, with 0 < t < 1, then f(C) > t f(A) + (1-t) f(B).) (c)* Extend f to a function g defined on all matrices by putting g(T) = (1/2) f(T*T). Is g concave? (d)*** Show that if h is any concave function, then Tr(h(A)) is also concave in A for all A Hermitian. Hint for (b): you may wish to use a characterization of concavity in terms of the Hessian.
  4. F0109; S0211; S0308; F0408; S0409; F0507; S0503; F0608*; F0612; S0608; S0704.