Recent content by fishshoe

So do I need something like \begin{array}{ccc} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \dots \\ 0 & 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & \dots & 0 \end{array} as an n-vector Jordan basis for the polynomials of order up to n?

Homework Statement Let V = P_n(\textbf{F}) . Prove the differential operator D is nilpotent and find a Jordan basis. Homework Equations D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1} The Attempt at a Solution I already did the proof of D being nilpotent, which was easy. But we haven't covered...

Yeah - I hadn't thought about using that for a proof, but it would show that (-1)^{n}T^n = 0 \Rightarrow T^n = 0 so T is nilpotent. So what does this part (b) mean? T^n = 0 means T is nilpotent no matter if the field is complex or reals, right? Is it just a mean-hearted distraction...

Hey, I'm working on a proof for a research-related assignment. I posted it under homework, but it's a little abstract and I was hoping someone on this forum might have some advice: Homework Statement Suppose T:V \rightarrow V has characteristic polynomial p_{T}(t) = (-1)^{n}t^n. (a) Are...

Okay, I figured it out. Nevermind (although once a post gets buried two screens back, it's not likely to be answered anyway, even if it has zero replies...).

I'm trying to prove that every nxn matrix can be written as a linear combination of matrices in GL(n,F). I know all matrices in GL(n,F) are invertible and hence have linearly independent columns and rows. I was thinking perhaps there is something about the joint bases for the n-dimensional...

So if I have A = diag(a_1,...,a_n), then A\vec{e_1} = a_1\vec{e_1} A\vec{e_2} = a_2\vec{e_2} ... A\vec{e_n} = a_n\vec{e_n} But a vector of all 1's should also be an eigenvector of A. A * (1,1,...,1)^T = (a_1, a_2, ..., a_n)^T And therefore this can only be an eigenvector if...

Also, the version of Bayes' Theorem you're using is more complex than you need to solve the first problem. The basic formula for conditional probability is P(B|A) = P(B \cap A)/P(A) So you know you're in A because that's given. So you want to know, what's the probability that you'll be in...

So if A is a diagonal matrix in any bases β and γ, then [A]_β = diag(a_1,..., a_n) and [A]_γ = diag(b_1,..., b_n) And for the eigenvectors in any basis, [A]_βe_i = a_ie_i But I'm stuck there. How do I show that a_1 = a_2 = ... = a_n ?

When you say that P(AB) = P(A) * P(B), you're assuming that the two events are mutually exclusive. But since P(A|B) = 0.7, we know that's not true (otherwise P(A|B) = P(A)). What you want is the probability of A and B (i.e. their intersection).

So since Av is in the same one-dimensional subspace as v, we know that Av is a scalar multiple of v, and so A is a scalar nxn matrix! But does this apply to any nxn matrix B? Or does it have something to do with the specific B that we defined, such that we have to generalize it further to prove...

I'm trying to figure out what v, b2, b3,..., bn is a basis for. Is it for all nxn matrices? If Bv=v, then v is an eigenvector of B corresponding to eigenvalue λ=1, and B is the identity operator on the one-dimensional subspace spanned by v. I know that det(B-I) = 0, so maybe something...

It's not clear where your brackets end in Bayes's Theorem. Try this instead: P(B|A) = P(A|B) *(P(B))/(P(A))

Homework Statement Prove: If a matrix A commutes with all matrices B \in M_{nxn}(F), then A must be scalar - i.e., A=diag.(λ,...,λ), for some λ \in F. Homework Equations If two nxn matrices A and B commute, then AB=BA. The Attempt at a Solution I understand that if A is scalar, it...

Homework Statement Prove: Every nxn matrix can be written as a linear combination of matrices in GL(n,F). Homework Equations GL(n,F) = the set of all nxn invertible matrices over the field F together with the operation of matrix multiplication. The Attempt at a Solution I know all...