2007 Boot Camp, Problem Set 1
One of the primary goals of boot camp will be to familiarize you with some techniques for solving the sorts
of problems that will occur in your 500/600-level classes. These same sorts of problems will occur on your
qualifying exams, so it’s good to get comfortable with them early, and they really are helpful for learning to
do research in the end.
There are four essential exercise types that occur in mathematics courses: define/state, true/false, com-
pute, and prove: Define/state problems require some level of memorization when one is new to a subject,
but rapidly become trivial as one’s understanding of why a particular definition is made or theorem is true
increases. Our purpose with these problems will be two-fold: to give you an idea which common definitions
and theorems you need to review and to try to make the reason some of these definitions and theorems are
the way they are clearer.
True/false problems tend to be the most difficult of the problem types when one is unfamiliar with a
subject as they test the sharpness of one’s knowledge – a slightly weakened condition makes something that
seemed true become false, or something that seems too good to be true if one isn’t thinking cleverly turns
out to be so. Our purpose with these problems will be to get you into the habit of writing down examples
for theorems you run across and counterexamples for those same theorems when conditions are relaxed. If
I had one habit that I could have instilled in myself before starting classes, this would be it. To answer
true/false problems: if true, write down an example and a sketch of the proof if you can; if false, write down
two non-trivial counterexamples. (Here, non-trivial means ”not the first thing that comes to mind.”)
Computations ground us back to reality. It’s going to be very tempting to forget that all of the theorems
we spend so much time proving exist so we can actually do computations. Taking a moment to apply the
theorems gives an opportunity to tie all of our knowledge back together and see it as a whole.
And lastly, proofs are fairly self-explanatory, as we all hope to be doing them for a living some day.
This first problem set is going to focus on Linear Algebra, though will contain a few basic problems of
the type you might see at the beginning of 500-level courses, just to spice things up. Linear Algebra is
generally assumed to be background knowledge by 500/600-level instructors at a level most undergraduates
don’t necessarily see or recall. Take some time to work through these problems on your own and we will
proceed depending on the average level of progress and number of questions when we get together for the
first time. Much of this will be used in the other problem sets and we will see quite a bit of other linear
algebra along the way. If you have questions, let me know!
1. Linear Algebra
Here, unless otherwise noted, Fis a field, Vand Ware Fvector spaces.
(1) Define...
(a) a field.
(b) an Fvector space.
(c) a linear map fbetween vector spaces.
(d) a basis for an Fvector space.
(e) similar matrices
(f) characteristic and minimal polynomials of a linear map
(g) eigenvalues and eigenvectors of linear maps
(h) diagonalizable linear maps
(i) Jordan Normal Form of a matrix
(2) Give as many examples of (types of) fields and vector spaces over those fields as you can.
(3) Let Vbe a finite dimensional vector space. Explain what it means for a matrix Ato represent a
linear map f:V→V. Prove that matrices Aand Brepresent fif and only if Aand Bare similar.
(4) Using the last problem, explain what is meant by a property of a linear map being basis invariant.
The basis invariant properties of a vector space/general mathematical object tend to be those that
matter.
(5) Let Vbe the set of all real polynomials in twith degree less than n (written by abuse of ring notation
as V=R[t]/(tn))
(a) Show that Van Rvector space.
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