I doubt it is necessary per se, it's kind of like asking "why did they rewrite x^2-5x+4 as (x-4)(x-1)". It's equivalent but sometimes one form is more useful than the other.

Although evaluating the phase at t=0 x=0 s=0 is helpful because it tells you what the phase is.

Okay. Humor me, what exactly is the "phase"? Is it simply the angle that the particle oscillates through?

Also, why was the displacement of the particle given as a cos function with a phase shift that indicates its the same as some other sin function? Why not just call it the sin funtion?

Phase is a tricky concept with a few distinct but related definitions. The wikipedia article on phase seems quite good and gives a better explanation that I think I can.

But phase isn't the angle a particle oscillates through. It is the angle by which one sinusoid is shifted compared to another.

Well if they just had it start as a sin function it's not as good an illustration of the concept of phase, is it?

That's totally reasonable, and I can appreciate that. I just wanted to make sure I wasn't missing something about why the manipulation might be preferable, or at least common out of some conventional necessity.

This Linear Algebra stuff is pretty cool, except for the amount of paper I need to work through any problem. There hasn't been anything earth-shattering to the level I've seen in Calculus, but I like the new way of thinking this course introduces.

Still early in the class and still learning about matrix inversions, triangle matrices, and transposing. If I'm understanding this correctly, I want to solve for A=LU (or PA=LU) to prove that one single point of the intersecting planes (or vectors) is a solution to the matrix. My question is, what is the practical consideration of solving for an invertible matrix? Why does it matter if a neat solution exists or not?

For some reason, I'm reminded of lowest common denominators from counting theory, though I don't quite have a complete connection to this yet.

Kind of going slow at first since I find matrix multiplication a bit difficult. I just break the matrices apart into vectors and do vector multiplications for right now, in the theory that the matrix multiplication will become more intuitive. Is this an okay way to get acclimated to the idea?

I hope I understand the question correctly. Often we want to solve linear systems like Ax=b; this appears in all sorts of applications. Doing this directly (via Gauss elimination for example) is quite complicated. If we want to solve several systems with the same matrix A, but different right hand sides b, then it pays to compute the inverse A^-1, since then for any b, the solution of Ax=b is just x=A^-1 b. Inverting A is a bit more expensive than just solving the equation once, but really just by a factor of 2, so if you have more than two b's it already pays off.

Another thing to note is that Ax=b is easier to solve if A is triangular; in fact it is of similar complexity as a matrix-vector product. And we can find a LU decomposition of A quite efficiently. If we know that LUx=b, we first solve Lz=b, and then Ux=z (both triangular). Since the decomposition is a bit easier to find than the inverse, this is practically a fast way to solve Ax=b for several b.

Yeah think Strang's course tends to start with a lot of matrix factorizations and manipulations, and I find all that kind of tedious.

Matrix multiplication is just a whole bunch of vector multiplications (dot products), so you're doing fine. And yes, these early factorizations, etc. are slow and take a lot of work by hand (until you learn to make a computer do them, but for now you should understand the mechanics of doing it "the long way").

And for the most part, whenever you are asking for the solution to a problem, it's nice to know whether the solution is unique or if there are many. For example, if there are many, you may want to further select a solution that is optimal in some way (e.g., costs the least).

I'm working on this problem where I need to get from the line above 2.3.17 to 2.3.17 (it took some work even to get to that line above ). I know that the integrals should be taken over the Gaussian distribution of y so that the integrals reduce to 1. So the main part seems to be expanding the norms. I've tried doing this for the 1st few terms.

It seems like the terms that involve y and x need to = 0? But I thought that would only be true if orthogonal?

Also seems like the pi*N_o term needs some manipulation to add a 2 to the denominator in order to have the Gaussian distribution for the integral.

Any tips? Thanks

Small correction, the variance is N_o/2 so the term outside the integral in the line above 2.3.17 is fine. Also my idea of expanding the norms and ignoring mixed y,x terms seems fine for y but missing a 1/2 fraction around the x_m - x_m' norm.

Chisness,

All you need to do is complete the square basically for the Gaussian integration;

ie try to write ;

(y-xm')^2+(y-xm)^2=2*(y-A)^2+B and find what A and B have to be.

You will find out that B is basically what you wanted to recreate ie a term proportional to (xm'-xm)^2. A also doesnt change the integration as it is only a translation (eliminated by say change of y variables to say y') but B drops out as a factor you want etc.

Obviously generalize to the norm for a vector in N dimensions to get the general result.


Also for other such problems recall the way to do the general Gaussian integral

http://en.wikipedia.org/wiki/Gaussian_integral

under the section "n-dimensional and functional generalization".

My signals professor had a pretty good line today: "If you're going to graduate as an EE, it's important that you have a pretty good understanding of high school math." I think that he is disappoint with some of my classmates.

"If the {sequnce s_n} is convergent and lim n->infinity s_n=s exists as a real number, then the series Σa_n is called convergent..."

There aren't two conditions there, right? If the sequences is convergent then the lim n-> s_n=s exists. I just want to make sure I'm understanding the relationship.

What is the significance of using "i" index when talking about sequences but then using n when talking about series?

"exists as a real number": What else should it exist as? Some people say that the limit exists even if it is infinite, but that is really sloppy. So you are right, this is just one condition. A series is called convergent if the sequence of partial sums converges.

(You could argue that it is convergent even if the limit exists as a complex number...)

None whatsoever, except for the author's sloppiness. One reason may be that you need another index to define the partial sums, i.e.,
.
What textbook are you using?

p-adic number.

Thanks. I'm using James Stewart Calculus and when he gives the definition of a sum he uses different notation for the partial sum and for the sum of the series. I understand the difference between those two concepts but just didn't know if there was a reason, other than to make clear the distinction between the two, why the notation was like that.

for the Harmonic series I'm told that, in general, S_(2^n) > 1 + (n/2). I'm wondering how this can be stated when it's not the case when n=1.

If I'm not mistaken it's true when n>1, but for the case where n=1 then S_(2^n) = 1 + (n/2).

That's also not a harmonic series, it's a geometric series. The harmonic series is 1/n:

1 + 1/2 + 1/3 + 1/4 + ...

which diverges. However the alternating version

1 - 1/2 + 1/3 - 1/4 + ... = ln(2).


Did you give up on my McNuggets problem? I told you exactly how to do it. There are 3 types of numbers: multiples of 3, 1 more than multiples of 3, and 2 more than multiples of 3. The 6 and 9 piece alone cover all the multiples of 3 except 3, and only multiples of 3. Now consider how to cover the other numbers using 20s, and find the biggest that can't be covered.

I can see how that could be confusing. It is the Harmonic Series I'm talking about, but here is a picture to clarify what I am asking.



Didn't give up yet, I've got midterms going on though so I haven't been able to do any pleasure problems

OK, the index is 2^n. It's only true for n > 1, but it only matters what happens when n -> infinity. It's bigger than something that diverges, so it diverges. He could have said >= to cover all cases, and it would still prove divergence.

More sloppiness.

Generally speaking, how do I know if I want to find a confidence interval for a population proportion or a population mean?

Are there key words to look for? Certain variable types?

Think I got it sorted. Ignore previous post.

Quick math related question. Say I play 3 tournaments. Between the 3 tournaments, I have a 15%, 25%, and a 35% chance to win prize (x) in each tournament respectively.

How do I calculate my chance to win prize (x) at least once after all three tournaments, assuming I played each one? I know it isn't as simple as (.15+.25+.35), but I cant remember the extra step I'm missing.

The simplest way is to calculate the probability of winning exactly none of the tournaments and then subtracting that from one:

0.85*0.75*0.65=0.414375

0.414 is the probability of not winning any, so the probability of winning at least one is 0.586

Alternatively you could add the chances of winning exactly all three, exactly the first two, exactly the last two, ....