Exercises for February 25th

Exercises for Monday February 25th: Chapter 2, problems 33, 34; Chapter 3, problems 3, 4, 6, 7; MA 252 exam 1997, problem 1.

Please work on the problems in advance, and come to the class with an idea of which problems you would like to see discussed. We will only have time to go through a few of the problems.

Hints: For problem 33, to show that 1 is a regular value of the determinant map from M(n, R) to R, it suffices to consider the curve c(t) = tA through any matrix A of determinant 1, for t near 1. This replaces the explicit formula for the derivative of the determinant. For problem 3, note that the map GL(n, R) --> GL(n, R) taking an invertible matrix A to its inverse is continuous. For problem 4 either use that in a vector bundle E --> B the space B has the quotient topology from E, or use the zero section to think of B as a subspace of E. For problem 6 make use of problem 34 from chapter 2. For problem 7 remember that to give an ordered basis for an n-dimensional vector space is the same as choosing a linear isomorphism with R^n.