Recent content by brian44

Ah, thanks! I didn't even think of using the mean-value theorem...

We may assume f and f' are continuous, since that is not the main question I am concerned with. It is not quite the same thing, since that is not how the derivative is defined - all definitions I've seen have one value fixed. I.e. just plugging in x_n to f'(x) does not give the same...

Given a convergent sequence x_n \rightarrow x and a function f, is lim_{n \rightarrow \infty} \frac{f(x_n) - f(x_{n-1})}{x_n - x_{n-1}} = f'(x) ? I believe it it is, but I haven't been able to figure out how to prove it. Does anyone know of a proof or counter-example? And probably should...

To show it is a linear function of the data it is enough to show it is a matrix multiple of the data, i.e. {B0;B1} = GY where G is a matrix, since matrices are linear operators. Specifically proving a transformation is linear means showing T(aX+b) = aT(X)+b where a and b are scalars scalars...

Assume there are k tracks, giving a total of k+1 choices (player may choose random). Assume the computer selects one of the tracks, or random, with probability according to the number of votes/total for each, call the number of votes for track i v_i, i =1,..,k, number of votes for random v_r...

The problem with such an approach is that it uses the exponential function and its properties in the definition of the exponential function itself - so I would consider it circular reasoning. Before defining e^x we have definitions for integer powers as products, but without defining the...

In the limit as w \rightarrow \infty I believe they are always equal. I will use the probability density functions (f(x),f(y), and f(x,y)) to give my reasoning. P(wx < c) = P(x < c/w) = P(x < 0) in the limit of w \rightarrow \infty = \int_{-\infty}^{0}f(x)dx =...

I've tried and searched for a long time, and I haven't been able to prove or find a proof that the following sequence converges (without using another definition of the exponential function): \forall x \in \mathbb{R}. Prove that: \lim_{n \rightarrow \infty} (1+ x/n)^n exists. I can...

Thanks I see now how the first part is obvious: Even if we don't fix the alphabet, the set of all finite automata is a countable union of at most countable sets and hence is at most countable. Therefore the set of regular languages, those that can be recognized by some finite automata, is at...

I was wondering, since the size of complementary sets comes up often in other areas of math, e.g. the set of rational numbers is countably infinite but set of irrationals is uncountably infinite, so that the set of irrationals is in some sense "larger" than rationals. Does a similar kind of...

You are correct, the range of B has the same dimension, however, B is not a linear operator, because a linear operator is a linear transformation from a space to itself, i.e. the same space, it must take P_4 -> P_4, but tA(t) takes P_4->P_5, although to a 2 dimensional subspace in P_5 it is no...

This seems incorrect to me, it should be M. The reason is we usually capture the covariance matrix using X*X' where X is an NxM matrix of N features by M observations (usually we normalize X so that each row [feature] has mean 0 and variance 1, or some similar process, so that we are getting...

I would also put Rank + Nullity Theorem at the top, or the RCF theorem (see below). Perhaps we have been following a bit different set of topics than in your course, but for us one of the most important theorems was the Rational Canonical Form Theorem, and subsequently Jordan Canonical Form -...

He means a basis for the kernel and a basis for the range. So you have found the range to be all of R^2, then you may give any basis for R^2, recall a basis is a set of vectors from the space such that they are linearly independent and span the space. There is a standard basis for R^2 you...

Actually as someone suggested, it is very easy using the definition of e^x as the infinite power series: \displaystyle\sum_{k=0}^\infty \frac{x^k}{k!} . Try taking the derivative and you will see it is the same sum, so that D(e^x) = e^x. Then you may use L'hopital's rule, whose derivation...