Recent content by Bipolarity

I'm not really sure if this is true, which is why I want your opinion. I have been trying to prove it, but it will help me a lot if someone can confirm this. Let ## v_{1}, v_{2} ... v_{n} ## be vectors in a complex inner product space ##V##. Suppose that ## | v_{1} + v_{2} +...+ v_{n}| =...

I see! Thank you all for your replies! I knew I had seen it somewhere, little did I know it was right there in the proof of the C-S inequality itself! BiP

Try using this with what you have: If the product of two linear maps is invertible, then each of those maps is invertible. BiP

Conjugate symmetry, linearity in the first argument, and positive-definiteness.

Proving this should not require the definition of the inner product, only the properties.

I'm learning about Support Vector Machines and would like to recap on some basic linear algebra. More specifically, I'm trying to prove the following, which I'm pretty sure is true: Let ##v1## and ##v2## be two vectors in an inner product space over ##\mathbb{C}##. Suppose that ## \langle v1 ...

Ah I see, I suppose then that each number has half the probability of its successor? Also, do you think there might be a way for me to tweak this so that the numbers are indeed equaprobable? This is for an application I'm trying to build. BiP

Suppose you have a list of numbers, say ##{1, 7, 9, 4, 5, 6}##. You store the first number, and then iterate through this list. For each number in the list, you flip a coin. If it is heads, you swap that element in the list with the number you stored. If tails, you do nothing. Either way, you...

Suppose you have a distribution ##p(x, \mu)##. You take a sample of n points ## (x_{1}...x_{n})## from independent and identical distributions of ##p(x, \mu)##. The maximum likelihood estimator (MLE) for the mean ## \mu ## is the value of ## \mu ## that maximizes the joint distribution ##...

P.S. Slight mistake, I meant to prove that surjection implies injection, not the other way around. I agree with the definition that two sets have equal cardinality if a bijection exists between them. But how does that prove that every surjection from ## S ## to ## T ## is also a bijection? So...

This is a rather simple question, so it has been rattling my brain recently. Consider a surjective map ## f : S \rightarrow T ## where both ## S ## and ## T ## are finite sets of equal cardinality. Then is ## f ## necessarily injective? I proved the converse, which turned out to be quite...

Suppose F is a field and that ## f(x) ## is a non-constant polynomial in ##F[x]##. Since ##F[x] ## is a unique factorization domain, ## f(x) ## has an irreducible factor, ## p(x) ##. Then the fundamental theorem of field theory says that the field ## E = F[x]/<p(x)> ## contains a zero of ## f(x)...

In ring theory, a polynomial over a rings, say ## R[x] ## is presented as an abstract object of the form: ## p(x) = a_{n}x^{n} + ...+ a_{1}x + a_{0} ## where the coefficients ## a_{n}...a_{0} ## are from a ring R with unity and ##x## is a formal symbol. So what is the significance of ## p(x+1)...

Let ##G## be a group and let ##R## be the set of reals. Consider the set ## R(G) = \{ f : G \rightarrow R \, | f(a) \neq 0 ## for finitely many ## a \in G \} ##. For ## f, g \in R(G) ##, define ## (f+g)(a) = f(a) + g(a) ## and ## (f * g)(a) = \sum_{b \in G} f(b)g(b^{-1}a) ##. Prove that ##...

The total probability of error is the sum of probabilities of type 1 and type 2 errors respectively. I am aware of the ROC curves, but that does not answer my question.