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#1 Help Me ! » Linear transformation from R^{3x3} to R » 2014-07-19 15:36:50

White_Owl
Replies: 1

We have a linear transformation from a matrix space, to a real space defined as a product of all elements in the matrix.
How can we define a matrix representing such transformation with respect to the standard basis?

The 'standard basis' here is a set of matrices:


But the result is not a matrix.

So I can write something like that:


But how to define a matrix representing such transformation?

#2 Help Me ! » Reflection over the line in r^3 » 2014-07-19 13:20:10

White_Owl
Replies: 0

Question:

is the reflection over the line spanned by (5,-4,7). Find a basis
so that the matrix representing T with respect to
is diagonal.


Here is what I understand:
1)
Since the matrix representing T is in "the most appropriate" basis and it is diagonal, it have to be


Correct?

2)

where
and

Correct?

So how do I find b2 an b3?

#3 Re: Help Me ! » Invertible matrix » 2014-07-14 07:54:51

Vectors which undergoing the transformation.

#4 Re: Help Me ! » Invertible matrix » 2014-07-14 06:38:05


hmm... Ok. It does look correct.

The next question is what if A is not invertible? Does it affect invertibility of T or not?

#5 Re: Help Me ! » Invertible matrix » 2014-07-13 14:38:11

Oops. mistake.


So my nice set of equations is not correct sad

#6 Re: Help Me ! » Invertible matrix » 2014-07-13 14:27:57

Homomorphism? I guess not. At least what I read in my lecture notes does not give me any clues on how to solve the problem.

Ok. Taking a step back. According to my notes:
- Let T:V->W be a linear transformation.
- If T is invertible, we call it an isomorphism.

So to prove that

is isomorphic, I need to show that it is invertible. Yes?
And if T is invertible, that  would mean that
has a unique solution.

Well, maybe the trick is in the properties of invertible matrices?


It does look kinda nice...
Does that work as a proof, or did I prove something else?

#7 Help Me ! » Invertible matrix » 2014-07-13 10:52:30

White_Owl
Replies: 8

Problem:
Let

be a square invertible matrix. Let
be the linear transformation defined by
. Prove that
is an isomorphism.

How to start the proof?

I am thinking about something like this: Let

, then by definition of matrix invertibility
. So
.

But there is an issue of order, it does matter in the matrix product and I do violate it in this proof. So it must be wrong. But I do not see how to fix it.

#8 Re: Help Me ! » Find volume of intersection of two spheres » 2014-03-18 04:54:39

bob bundy,
you mean to draw a line through centers of the spheres, and look at this line as at a 'new' x-axis? Yes, this will work, thank you.

#9 Help Me ! » Find volume of intersection of two spheres » 2014-03-16 05:38:30

White_Owl
Replies: 3

We have two spheres with centers (x_1, y_1, z_1) and (x_2, y_2, z_2). And radii r_1 and r_2.
What is the volume of solid intersection of these two spheres?

I am not sure how to approach this question.
I know that a point will belong to the solid iff:


But how to find a volume limited by this surface?
It is not a curve, so I cannot use  a straight 'solid of revolution' formulas, can I?

It should be something like a sum of all points which satisfy the inequality:


but how?

#10 Re: Help Me ! » Number of changes » 2013-10-20 11:18:37

After a second look at all these... I have a feeling that on average, we would have exactly half of the bits inverted. So we can say that if we increase number of bits, on average, the randomized inversion will require half the power than full inversion.
Is this correct or am I oversimplifying it?

#11 Help Me ! » Number of changes » 2013-10-20 11:13:13

White_Owl
Replies: 2

We have two  three digit binary numbers. In one number all digits do invert every single step (does not matter what the initial value was, lets say we have a 000-111).
In another number, each digit inverts on random.
Each inversion requires some work to be done. So for the first number, in 10 steps we require 3*10=30 energy.
If none of the digits in the second number was inverted (this can happen), then in the same 10 steps we require 0*10=0 energy. If all of the digits in the second number was inverted (this also can happen), then we will require 3*10=30 energy.
But since the second number is under randomizer we will usually have less energy spent on it in comparison with the first number.
The question is: what is the average reduction of energy which would be required by the second number in comparison with the first one?


My current train of thought is:

There are four possible cases:
0 inverted - 2^3 combinations
1 inverted - (2^3)^3 combinations
2 inverted - (2^3)^3 combinations
3 inverted - 2^3 combinations

At each step one of those four cases can happen.
So the probability to spend get any single from combination A to combination B is


We have
probability to spend 0 energy at one step
We have
probability to spend 1 energy at one step
same for 2 and 3.

So on average we supposed to get


In other words, the second number will spend 1.5 energy to invert random number of bits and on average we will spend 1.5*10=15 energy on the second number.

Is this correct?

#12 Help Me ! » How to find a number of trials? » 2013-04-07 13:28:30

White_Owl
Replies: 1

How to find n in the equation:


Here cdf_binomial() is a distribution function  of binomial random variable:

By trial and error I found that n=164. But is it possible to express n for a direct calculation? How did people solve such problems before the age of computers?

#14 Re: Help Me ! » Convergence of (n!)^2/(kn)! » 2013-03-31 11:28:04

anonimnystefy, thank you. I see the mistake now

So my new answer is:


Therefore, series converges for k>=2

#15 Re: Help Me ! » Convergence of (n!)^2/(kn)! » 2013-03-30 13:54:10

I do not think there are mistakes:


Or are you talking about different equations?

#16 Help Me ! » Convergence of (n!)^2/(kn)! » 2013-03-30 12:31:44

White_Owl
Replies: 10

The problem is:
For which positive integers k is the following series convergent?

My Answer:
For series to be convergent the next inequality should be true (by the Ratio Test):


Since we know that both k and n are positive we can omit absolute bars.

And now I simplify:

But since k is a constant this limit will never be less than 1. Therefore the series divergent for all possible k.

Did I make a mistake somewhere? Textbook is looking for a convergent series...

#17 Re: Help Me ! » Estimating a sum of a series » 2013-03-17 14:55:25

Oh.... I see it know.
Actually the approach you used is discussed in calculus - the Squeeze Theorem. I just did not think it can be used here as well.
Thank you.

#18 Re: Help Me ! » Estimating a sum of a series » 2013-03-17 12:12:47

Could you please elaborate: what other means?
I am reading Stewart's Calculus Early Transcendentals (7th edition) and there is only one method described (ch 11.3)

#19 Help Me ! » Estimating a sum of a series » 2013-03-17 10:55:35

White_Owl
Replies: 5

The problem: Calculate a partial sum for the first ten terms, estimate the error.  Round answers to five decimal places.

Well, the partial sum is not a problem - a few minutes with a calculator and the answer is 1.04931. But the error estimation part gives me the trouble.
As far as I understand, I am supposed to solve:


And to solve this I need to find an integral of that scary function. But I have no idea how to approach this integral.

#20 Help Me ! » Summation of harmonic sequence » 2013-03-07 02:33:30

White_Owl
Replies: 1

Problem:


Here c is a positive constant, find is this sequence convergent or divergent and if it is convergent, what it converges to?

I see that the series is convegent.

And I also see that it converges to some value which is bigger than 1/c, but less than 2/c.
But what exactly is the value? How to reach it?

#21 Re: Help Me ! » Expected value of a function of a random variable » 2013-03-06 08:58:49

Ok, I tried it both ways and got:

if h(x)=1.5-x, then


if h(x)=x-1.5, then

Maybe I do not understand it at all, but: If pdf is defined for the [1,2] range, does that mean that we expect "the demand" to be at least 1, but no more than 2? but from this point of view the 1.5 at the beginning of the week will satisfy part of the demand and at the end of the week we would have a shortage. So negative amount should be the correct answer?

#22 Help Me ! » Expected value of a function of a random variable » 2013-03-06 08:44:55

White_Owl
Replies: 2

The problem states:

The weekly demand for propane gas (in 1000s of gallons) from a particular facility is an rv X with pdf


If 1.5 gallons are in stock at the beginning of the week and no new supply is due in during the week, how much of the 1.5 thousand gallons is expected to be left at the end of the week? [hint: let h(x)=amount left when demand =x.]

As far as I understand, to solve the problem I need to calculate:


And E(h(X)) would be the answer for the problem. But what is the h(x)??? Is it (1.5-x) or is it something else?

#23 Re: Help Me ! » Gabriel's Horn » 2013-03-03 17:11:13

Divergence should be proven or shown...
I think this solution can be used as a prove, but maybe there is an easier way?

And I repeat the question: Why Wikipedia uses incomplete formula?

#24 Re: Help Me ! » Gabriel's Horn » 2013-03-03 12:06:17

Another attempt:
Starting from here:


Let

Then:

We have a formula #24 in the table of integrals in the textbook:


So:

And here we have first limit is infinity divided by infinity, second limit is infinity and a constant.
Therefore we have an infinity in the final answer...
Did I make any mistakes?

#25 Re: Help Me ! » Gabriel's Horn » 2013-03-03 11:36:25


Now I wonder myself where did I got u^6 smile Thanks.


Now I am not sure what to do next?

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