I'm helping a friend with Linear Algebra. I never took the course as an undergraduate, so some of the terminology is kind of lost on me. Anyways, I googled some things and have an idea.
The problem states:
Of course I know what a transpose of a matrix is.

The null(T) is just the Kernel. im(T) is the image of T, also known as a range for functions (what the map maps onto).

So, the general description that I've found is that you find a linearly independent set of nxn matrices (Most easily explain the matrix with one element in each position, but possibly something special like the Pauli Spin Matrices) where there will be n^2 of them and use the linear map to obtain a new set of matrices. At this point I'm confused. I know that n^2 is the number of elements in null(T) and im(T) combined.

So, for 2x2 matrices, you can use the Pauli matrices (with the identity of course) and you see that one of them maps onto the matrix of zeros. This is in the null(T) and the other elements are non-vanishing and thus are in the im(T)?

You can't just take any basis for the entire matrix space then apply T to it, it would have no reason to contain subfamilies that are bases for null(T) and Im(T).

Also you probably know this but n^2 is not the number of elements of these subspaces but the sum of their dimension (elements in their bases).

Here it is easy to see what both of these subspaces are (hint : find a relation between X and Xtr for X in null(T) and Im(T)), and finding a base is not that hard either (write these relations as elementary equalities of the type x_{i,j] = x_{k,l}, you can first look at n=2 to get the general idea).

Thanks for the response check.

Yes, of course.

The relationship for null(T) is X = Xtr (ie, all elements are the same). Then the image(T) would be all matrices X != Xtr.

The basis of both of these subspaces must be linearly independent in the original X space, right? So the basis for null(T) is composed of (n^2-n)/2 elements; that is, elements with 0s in the diagonals and say 1 at x_{i,j} and -1 at x_{j,i} such that i != j. That is, for a 3x3 matrix
0 a b
-a 0 c
-b -c 0

where the basis matrices in the subspace are just the aX_1, bX_2, cX_3. (9-3)/2 = 3 and that's how many there are.

Similarly, the basis for im(T) are all matricies is a set such that x_{i,j}+x_{j,i} != 0. We know that there are (n^2+n)/2 bases for this set. So, all of them for a 3x3 matrix would be

c d e
d f g
e g h

Where they are cX_1, dX_2, eX_3, fX_4, gX_5, hX_6 which any combination gives the above. I think I understand.

No, X = -X^T, but you seem to know that below.
Im(T) = {X = Y + Y^T: Y in M(n)}. Note that Im(T) is contained in the symmetric matrices. Then show that all symmetric matrices are in Im(T). Then find a basis (you did) and show that it has the appropriate size.

Wyman already posted how to get the solution for this particular exercise so I won't get into it, but you seem to have some confusion here.

In general there is no simple relation between ker(T) and im(T) except for the formula about the sum of their dimensions, and certainly nothing like X \in Im(T) <=> X \notin Ker(T). It is a particular feature of this exercise that the space M(n) is the direct sum Ker(T) + Im(T) but this is not a general fact.

Re: the first integral:
I don't like your argument about a limit; it's not convincing. You should be more precise, though your conclusion is correct. An inequality would be easiest (and it is quite easy). In the divergent case, you should bound the integral below by something known to diverge. In the convergent case, you should bound the integral above by something known to converge.

Re: the second integral:
For simplicity, let p = -a/sigma^2, and let q = b/sigma^2. Assume q > 0.
Exponentials dominate polynomials, so for some large enough value X, we have the following:
x^p < e^(qx/2) for all x > X.
Now, then our integral converges iff the integral from X to infinity converges, but the integral from X to infinity is <= the integral of e^(qx/2)*e^(-qx) = Integral of e^(-qx/2), which converges.
So this converges when b > 0.
If b is negative (and thus e is being raised to a positive power),
then if p > 0, then for large enough x, both x^p and e^(-qx) are greater than 1. Since the integrand does not go to 0, the integral cannot converge.
If p < 0, then notice that for some large enough value X, we have:
e^(-qx) > x^(-p) for all x > X.
So again, the limit of the integrand (as x --> infinity) is > 0, so the integral cannot converge.

hth

Thanks to Wyman and CtR for both helping me in a pinch!

I've been struggling a bit with the game NIM. The game works like this: There are a number of piles of sticks. One player takes one of the piles and removes one or more sticks. The player that removes the last stick wins. Now to the problem.

1. Let's say that we have a game with 3 piles, for example 7 5 2 . Let's say that we have to make the first move. How do I determine whether I have a winning move or not? I calculate the Grundy value right (call it G). If G = 0 then I will lose no matter what but if is positive I have a winning strategy.

So for this game we have that (writing 7, 5 and 2 in binary)

7 = 111
5 = 101
2 = 010
_______
Gvalue 000 = 0 i.e. I will lose this game if I start.

But what if the question was this: Given that you have a particular game where if you start you have a winning move, what is the maximum amount of sticks you can take away and still win the game? How do I determine that number? The grundy value only tells me if I have a chance to win or not, it doesn't tell me anything about which moves I can make to win the game (max number I can remove and still win for example).

2. Assume I have an impartial game as follows



I don't know how to read this game, can someone explain? Also, how many positions does this game have?

If you make a move from initial position P_0 to P_1, what does the Grundy value of P_1 tell you ?

I think this notation means :

You start with a set X_0. Player 1 chooses a subset X_1 of X. Then Player 2 chooses a subset X_2 of X_1, etc... until you reach the empty set, at which point the player whose turn it is to play loses.

Wyman, thank you for your help!

Thanks. I solved the first problem but are still having problem with the G = ...

For example, can I simplify the expression? Isn't {{empty set}} = {empty set} = empty set? When you say a set X_0, do you mean the empty set?

Hey guys, I'm trying to implement a simple 9-point stencil Finite Difference method in matlab, but I'm getting too large a truncation error. I think it's likely a one or two line problem (since I got the 5 point stencil to work perfectly). If any of you guys would be interested in helping me out, let me know!

Thanks!

This question is from hw 1 of a financial math/stat course. I'll assume the math is pedestrian here, but I haven't had to apply calculus in almost 10 years, so im lost.

What the distibution says is, essentially, no one in the sample population lives to 90, thus the first expression integrated I would imagine has to be 1.
If this is the case, I view C as per the first question as a weight that will account for this. I.E. 1/C = integral e^-0.02x.
problem is that I really dont know how to integrate. The best I can piece together from memory is e^anything integrates to itself, and that quantity is multiplied by the integral of the exponent. Thus, what I think I have:

1 = C* | [-0.01(x)^2]*e^-0.02x (where | is my drawing of the integral sign with bounds 0 and 90)

or C = "above integral"^-1

evaluated for 90 yields C = [[-0.01(90)^2]*e^-0.02(90)]-1 and evaluated for 0 is 0, so the integral over the range [0, 90> yields a C of -1.4.

I feel so lost, please help

Integrate from x = 0 to x=90.

Do a u-substitution.

u=-.02x
du = ?

Then change the limits of integration.

To normalize your function, you set the integral = 1 and you can say C*integral = 1 to solve for C.

wikipedia states the integral of e^cx dx = the integral of 1/c * e^cx http://en.wikipedia.org/wiki/List_of...tial_functions . If u = -0.02x, I would want to say du = -0.02. How do i reconcile that -0.02 =/= 1/c?

If you make a u-substitution, c=1.

Also, du = -0.02dx.

I think I got this one now. I wasn't solving du = -0.02dx for dx and using that. in e^u*du.
Thank you for your help/patience.

Noting e = emptyset. {e} is a set with one element (the empty set), and it's different from e which is a set with 0 elements.

In the example you gave, X_0 = { {{e}}, {e,{e}} }, a set with 2 elements (and these elements are also sets). Player 1 chooses one of these elements (I said subset in my earlier post and that was wrong), then the game continues with the new set, etc..until one player can't choose anymore because we have reached the empty set.
For instance :
P1 chooses {e,{e}}.
P2 chooses e.
P1 loses.

or

P1 chooses {{e}}
P2 chooses {e} (not really a choice here)
P1 chooses e
P2 loses.

Okey,

I have done some progess...

G = { {{e}} , {e, {e}} }

= { { *1} , {*0 , *1} }

= { { *1} , *2 }

but I don't know how to simplify that last part.

I have used that *n represents a game of Nim with a single pile of n sticks. Hence we know that the game *1 has only one option, *0, i.e. *1 = {*0} and the game *2 has two options *1 and *0, i.e. *2 = {*1, *0}. Also we know that the game *0 has no options i.e. *0 = e = {}.

Looking at { { *1} , *2 }

can we see this as a NIM game with three piles with *1, *1 and *2?

ship it

i am taking graduate probability as a senior at uiuc. my abstract measure theory is a bit shaky. i think this problem is straightforward but i am not sure what i need to prove and how to prove it. i could use a starting point or a theorem or technique to start out. i have shown u is a probability measure already, so it's down to proving the expectation relationship. i know the integral exists since phi is bounded and measurable. thanks for the help. i appreciate it.

Let X be an S-valued random variable. define u(A)= P{x in A} for A in L
Show u is a probability measure on (S,L) and that for any bounded measurable
phi:S->R, E[phi(X)] = Integral{(phi(z)u(dz)} and the integral is over S.

I'll try and work on it tonight, but if nothing works, I'll PM you. Thanks for the help offer!

You generally want to do these for a simpler class of functions first. For example, if \phi is the indicator function of a measurable set, then its trivial. Approximate a general _positive_ bounded measurable function by a linear combination of indicator functions of measurable sets, and use monotone convergence. For arbitrary bounded measurable functions, split into positive and negative parts.

I have a game theory/mixed strategy question. If anyone knows how to figure this out I would greatly appreciate it! (our professor hasn't discussed solving n-player games in class much)

At a party, there is a group of N friends who want to talk with some man at the event. All of them most prefer talking to Ben. There is time for one conversation. Payoffs are the same for all friends. If one talks to Ben alone, the received payoff is 2. If they talk to any other man alone, the payoff is 1. If they talk to man together with another friend, the payoff is 0.
Consider the following strategies of the N friends: "talk to Ben", "talk to one of the other men who is alone" and "talk to no one."
Note that the friends choose simultaneously and there is no going back from their decision.
a) Suppose N= 3. Find the symmetric MSNE
b) Suppose N= 4. What is the MSNE? What happens to the probability of a friend talking to Ben as N increases?

Given the wording of the problem, I am pretty sure we can consider the strategy "talk to no one" as being strictly dominated since at worst, a friend could talk to another man (besides Ben) and receive a payoff of 1 instead of 0 for not talking at all.

If anyone can come up with a solution I would greatly appreciate it!