Recent content by calorimetry

Homework Statement This problem introduces a simple meteorological model, more complicated versions of which have been proposed in the meteorological literature. Consider a sequence of days and let Ri denote the event that it rains on day i . Suppose that P(Ri | Ri−1) = α and P(Rci | Rci−1c) =...

How do you know that the equation xA = Bx has a complex solution? And what does that have to do with PBP^-1? I don't see the reasoning, care to elaborate?

I'm stuck on 4b, you got any plan on how to do it?

Eh not sure what you just said, but I just make P to be the identity matrix, then C is the same as the matrix between P and P^1. After all, it asks for some matrix P.

You can't define V to be span of v for case 2 because V is a subspace in R^n and v is a complex eigenvector (i.e. the span of a complex vector will create complex vectors outside of R^n). Try separate v into Re(v) and Im(v) and make V to be a span of those vectors. (This is where dim = 2 come...

Remember V can be almost anything that you want as long as you define it properly and it satisfies the given conditions. What I did was separating the problem into two cases, the first one is pretty easy, the second one, not so much. Case 1: A has some real eigenvalue Then there is some...

What if A is real, but it does not have any real eigenvalue (i.e. only complex eigenvalues) like A = [0 -1],[1, 0] a 2x2 matrix? I don't think that your line of argument would apply here since there is no eigenspace in R^n with complex eigenvalues. :(

Ah thank you, it's so simple when you put it like that. :D

Homework Statement a)Let A and B be nxn matrices such that AB=0. Prove that rank A + rank B <=n. b)Prove that if A is a singular nxn matrix, then for every k satifying rank A<=k<=n there exists an nxn matrix B such that AB=0 and rank A + rank B = k. Homework Equations rank A + dim Nul...

I guess the proof was simple after all. THANK YOU for your help :D

I guess what I learned is the same as kernel but defined differently (without using the term kernel). Theorem: Let T: R^n --> R^m be a linear transformation. Then T is one-to-one if and only if the equation T(x)=0 has only the trivial solution. Invertible Matrix Theorem: All statements are...

I guess column echelon form will work here, but I don't know if I am allowed to use column echelon form because we have not define that in class and the book hasn't mention it.

Pick two vectors in R^n v1 and v2 such that they have all the same entries except for the last entry and not all entries are zeroes. So v1 and v2 are linearly independent because they are not multiple of each other. Let E1, E2, ... , Ep be the series of row operation that reduce A into echelon...

(Ep...E2E1)^-1 is invertible (product of elementary matrices is invertible, theorem) so (Ep...E2E1)^-1 (a v1 + b v2) = 0 has only the trivial solution (by a theorem in my book). Hence, a v1 + b v2 = 0 is the only solution. I still don't see why v1 and v2 have to be linearly independent at...

Yes, this is part of the Invertible Matrix Theorem, which is already proved. Yes, there is also a theorem and proof on this. Sorry for not mentioning them. I'm a little confused here. I thought that it might not work for any two arbitrary vectors because of the way the problem stated "there...