Recent content by CoachZ

Hello all... First, let me say that I am currently a graduate student in mathematics who would like to change graduate programs to something involving geology. I've always been fascinated with the internal Earth and how it behaves. To me, this seems like a big jump. My questions are the...

If you were in Linear Algebra at a graduate level, or have been at some point in the past, what are the main theorems you would think of as being the most important for that course. In the next few days I have an exam and this is very good for my studying. Here's my top 14 list (nice round...

Haha, I woke up this morning thinking about I + iT and I - iT, and how they are both non-singular, which uses the theorem of real eigenvalues, and I was thinking that a + iTa = b + iTb probably uses similar concepts and ideas to solve. Thanks for the help!

Let V be a complex inner product space and T a self-adjoint linear operator on V. I'm trying to show a + iTa = b + iTb, iff a = b. The converse is trivial. The forward direction is getting me for some reason. Perhaps it's too late on a Friday night that my mind is completely gone. Any...

Figured this out. Since every operator takes one some matrix representation, and we consider the orthonormal basis with respect to the differentiation operator, we can extend that basis to matrix form. But there's a theorem that states the matrix representation of D* is simply the transpose of...

Exactly: D is the derivative operator and I'm trying to find D*, which is simply the adjoint operator. I apologize for writing it so messily. I'm a first year graduate student still working on my writing skills, so if you have any suggestions, either for finding D* or improving writing skills...

If the inner product is defined on V with dimension less than or equal to 3 as \left\langle f,g \right\rangle = \int_{0}^{1}f(x)g(x)dx, I'm trying to find D* such that \left\langle Df,g \right\rangle = \left\langle f,D*g \right\rangle, and I thought I had a closed form of D*. If {1, x, (x^2)/2...

I think that I'm a little confused, because if A^3 = I, then the eigenvalues for A would simply be the eigenvalues for I, which is just 1,1,1, since I is a diagonal matrix. Therefore, the characteristic polynomial of such A must be in the form (x-1)^3, right? Is there a way to compose matrices...

The question posed is "Classify up to similarity all 3 x 3 complex matrices A s.t. A^{3} = I. I think the biggest problem I'm having is understanding what exactly this is asking me to do. The part that says "Classify up to similarity" is really throwing me off, so if someone could tell me what...

Recently, I proved that Given f:A \rightarrow \mathbb R is uniformly continuous and (x_{n}) \subseteq A is a Cauchy Sequence, then f(x_{n}) is a Cauchy sequence, which really isn't too difficult a proof, however I'm having issues with the converse statement... More specifically, Suppose A...

Homework Statement Let V be the vector space of n x n matrices over the field F. Fix A \in V. Let T be the linear operator on V defined by T(B) = AB, for all B \in V. a). Show that the minimal polynomial for T equals the minimal polynomial for A. b) Find the matrix of T with respect...

Fekete's Lemma states that if {a_n} is a real sequence and a_(m + n) <= a_m + a_n, then one of the following two situations occurs: a.) {(a_n) / n} converges to its infimum as n approaches infinity b.) {(a_n) / n} diverges to - infinity. I'm trying to figure out a way to show either of these...

I'm trying to show that any uncountable set has a countable subset. First, let me point out that the distinction here between at most countable and countable is applied in this instance. At most countable implies either finite or countable, and countable is obvious. Starting off, let X =...

Let me clarify... Before I was a math major as an undergraduate, I was a pharmacy major, and I took two biology courses, namely Evolutionary Biology and Cell Biology and absolutely hated both courses. So, although my knowledge is fairly miniscule when it comes to biology, my experience with it...

So, I'm currently attending the University of Kansas as a Graduate Students in the combined M.A./Ph.D program. Essentially, we have the option of stopping once we get a Master's Degree if we choose, which is what I'm currently leaning towards at the moment. My question also falls under the...