Bootcamp 2007 - Linear Algebra

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*
  3. F0110; F0209; F0210; W0210; W0211; S0309; S0310; F0309*; S0410*; S049; F0611
  4. F0109; S0211; S0308; F0408; S0409; F0507; S0503; F0608*; F0612; S0608; S0704. Also do the following problem: (X1) State and prove the polar decomposition theorem for n×m (n≠m) matrices. (X2) Let T:V→W be a linear transformation. Show that the following are equivalent (a transformation with either of these two properties is called a partial isometry): (i) T*T and TT* are both projections (i.e., are idempotent, (T*T)2=T*T and similarly for TT*); (ii) let K=kerT. Then there is a finite-dimensional vector space U containing V and W, and a unitary Q:U→U so that if we set S equal to Q on the orthogonal complement of K in U and zero on K in U, then the restriction of S to V is T. (X3*) Prove that if V=W in (X2), one can take U=V, and in general, one can arrange that dimU=max(dim(V),dim(W)).
  5. S0701, S0702, S0705, F0605, F0606(a), S0607, S0608*, S0609, S0610, W0607, W0608, W0609, W0610. (X1) (a) Give an example of a function f which is discontinuous at every rational number, but is otherwise continuous. (b) Prove that f is Riemann-integrable.
For Monday 9/10 bring questions about problems F0506--F0510.
For Tuesday 9/11 bring questions about problems F0605--F0612.
For Wednesday 9/12 bring questions about problems S0701--S0705.
For Thursday 9/13 bring any remaining questions.
Special assignment (due Thursday, 9/20): Do all problems on F07.