Recent content by gucci1

Hey guys, I have a problem where I am supposed to prove that R is nonsingular iff A is of full column rank in a QR decomposition. I feel like I fully understand the two major processes for obtaining a QR decomposition (Gram-Schimdt and Householder Transformations), however, I am not entirely...

So I have been asked to prove a result that is supposedly valid for any norm on any vector space. The statement to prove is: | ||x|| - ||y|| | <= ||x - y|| The problem is, I have no idea where to start with this proof. Maybe I'm missing some fundamental property of norms, but it seems that...

Hi, for a homework problem I was asked to show that for an m x n matrix A, the l1-norm of the matrix is the max of the sums of the columns. What I tried to do was plug into the definition of the vector l1-norm with Ax for some x st the l1-norm of x is 1. This shows, after a bit of simplifying...

Thanks so much for the explanation, that really helped. I wasn't aware that it was so firmly established that the number of vectors in a basis is the same as the dimension of the space. I guess my teacher glossed over that fact :-/

I am currently trying to understand linear independence and spanning sets. So the question that I have right now is, does a set of five linearly independent vectors always span F^5? Thanks for any help you can offer! Edit: Sorry for the confusion, I really had no idea how to begin this...

So I get why everything after this step would follow, and thank you all very much for that. I am just missing why this is a true assumption :-/

The problem is: Let A be a real m x n matrix and let x be in R^n and y be in R^m (n and m dimensional real vector spaces, respectively). Show that the dot product of Ax with y equals the dot product of x with A^Ty (A^T is the transpose of A). The way I went about starting this problem is to...

Thank you so much! I love this forum, everyone here is so helpful :D

So, with your help I am much closer to the answer, but I'm making a mistake somewhere I guess. This is what I'm coming up with: 1/4 [|a+b|^2 - |a-b|^2] = 1/4 [⟨a+b,a+b⟩ - ⟨a+(-1)b,a+(-1)b⟩] = 1/4 [⟨a,a+b⟩ + ⟨b,a+b⟩ - ⟨a,a+(-1)b⟩ + ⟨b,a+(-1)b⟩] = 1/4 [⟨a,a⟩ + ⟨a,b⟩ + ⟨b,a⟩ + ⟨b,b⟩ - ⟨a,a⟩ +...

So, I have an equivalence I need to prove, but I think I'm having trouble understanding the problem at a basic level. The problem is to prove that the inner product of a and b equals 1/4[|a+b|^2 - |a-b|^2] (a, b in C^n or an n-dimensional vector space with complex elements). I don't...

Thanks so much man, that really cleared it up for me!

So I am having a bit of trouble with a proof by induction that I need to write. The problem is to prove that the conjugate of the product g1 * ... * gm equals the product of the conjugates of g1 ... gm. This is for g1 ... gm complex numbers. I have proven this for m = 2, by simple calculation...