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Suppose U is a finite dimensional vector space and A = {u1, u2, ... , un} is a basis of U. Define T : U -> R(nx1) by T(v) = [v]A.

(In words: U is an n-dimensional vector space, A is a basis for U, and T is the transformation that takes a vector in U and finds its coordinate vector with respect to the basis A.)

Show that T is a linear transformation.

My attempt:

If v and w are arbitrary vectors in U and a and b are scalars, we have

T(av+bw)

= [av+bw]A

=[av1+bw1, ... , avn+bwn]A

=[av1 + ... + avn]A + [bw1 + ... + bwn]A

=a[v1 + ... + vn]A + b[w1 + ... + wn]A

=aT(v) = bT(w)

I just don't see if what I'm doing here is justified. Please help! Thanks.

...that is, "Madrid".

I know it's difficult to convey pronunciation over the net, so... good luck!

Not sure if anyone on here will be able to help me with this...

I'm having trouble constructing a derivation for the following:

For all x, (Bx -> Ex).

There exists some y such that (Ay -> ~Ey).

Therefore, there exists some z such that (Az ^ ~Bz).

Apparently this is called "Baroco", if that helps. I've tried everything I know. Help!

i mean, compared to all the nice websites out there now, yahoo! answers is a pretty ugly website.

I use Bio-Oil for my stretch marks and scars. But I think it smells like old ladies. Is there any other product that works the same way, but doesn't have that disgusting lingering stench, or better yet, has no scent at all?

I've always wondered about this. Is it to make people sneeze more to expel all that mucus?

Basically, I am sick of Rogers, and from what I've read, Bell customers seem to be just as unhappy. See the following page:

http://www.ellenroseman.com/?p=446

Are there any other ISPs available in Toronto? I've also read that some "independent" ISPs just buy bandwidth off of Rogers or Bell and resell it, just giving you another incompetent rep to deal with before you speak to a manager about your complaint. Is there anything that can be done?

I really think the government needs to do something about the oligopoly that these two companies have on the telecom market.

I don't understand this experiment.

One version that I've read is that photons are allowed to enter the small hole and bounce around inside the cavity. They are then absorbed because they are trapped. The cavity reaches thermal equilibrium because the amount of energy it is absorbing from the photons is the same as the amount of energy that it is emitting. It emits energy as black body radiation. This radiation is at all frequencies, but the amount of radiation at each frequency is dependent upon the temperature. Please correct me if I'm wrong.

Another version I've heard is that there is an oven with a small hole in it that is being heated. The radiation will also be black body radiation. What is the difference between this "oven" and the previously mentioned "photon cavity"?

And I don't understand the "ultraviolet catastrophe". What does that mean? What does it mean to say that the energy at high frequencies will be infinite? What energy are we talking about at that point? The emitted black body radiation? Why is the amount of energy limited at high frequencies?

Thanks in advance.

Can someone explain this to me in simple terms? Here is what I understand so far:

There is a cavity with a small hole in it, into which photons can travel. This approximates a black body. The walls of the cavity will absorb the photons and reach thermal equilibrium when the amount of energy absorbed equals the amount of energy that is re-emitted. This re-emitted energy will be in the form of thermal radiation.

I don't understand why photons with a very high frequency would cause the radiation to approach infinity. I don't understand what is meant by "modes". What is meant by "the probability of occupying modes"? What does it mean for there to be "quantized modes"?

I'm not having very much luck with getting answers to these questions on YA. I have a ton more questions about this kind of stuff that I still don't really understand. Hopefully you can help me out.

Thanks in advance.

I am trying to understand the concept of CMB radiation.

When we say that the CMBR has a temperature of 2.7 K, is this because the photons have been spreading out across space and getting less and less dense? If we go back in time, then the photons would become more "dense" and the radiation as a whole would have more energy and thus have a higher "temperature". Does this make sense?

so I know that a photon of a certain wavelength / frequency will have a certain energy equal to Planck's constant multiplied by the frequency.

but i've read that radio waves could heat food if they are "intense". what does it mean for radio waves to be intense? because they will have very low energy, correct? if i wanted to fry an egg with radio waves, what would i have to do? bombard the egg with many many many photons of low energy?

"A body emits radiation at a given temperature and frequency exactly as well as it absorbs the same radiation."

I'm trying to understand this in the context of black body radiation.

I understand how a black body can absorb radiation in the form of heat, but how can it emit light with a certain temperature? I don't understand how we can say that light has temperature. I thought temperature was a measure of kinetic energy.

This is a follow-up to my earlier question:

http://answers.yahoo.com/question/index;_ylt=Al9C_...

Thanks in advance for your help.

i don't understand what is meant by an object being in "thermal equilibrium with the radiation field". does this mean that the amount of light it absorbs is re-emitted as heat, with conservation of energy?

and if it is re-emitted as heat, then where does the colour and light of the blackbody radiation come from?

thanks!

i was having lunch outside today near queen's park in toronto and i was feeding these little brown birds with bread. and then i remembered that feeding bread might be bad for birds because it's all processed. and so i thought, maybe i can feed them a banana, that's natural plant based food. and they really seemed to enjoy it. the birds were tiny, chirpy, and brown, some had black markings on them.

is it safe for the birds, or should i just not feed the birds at all?

also, what type of bird is this? (if you can make a guess, based on my vague description of them)

What are the benefits of being a vegetarian? Not just benefits for myself, but benefits for the environment? I've heard that if everyone was a vegetarian, we would only need 5% of the land for food production, since cows eat so frickin much. and i guess that would help with the greenhouse gas problem.

if you can, just a concise list of benefits would be good. i ain't gonna read an essay.

thanks!

I'm trying to rent a house for September when I go back to school, and I just checked out a house today that I really like. I realized later that I did not see the bathroom. I don't even recall seeing a door for the bathroom, otherwise I would have looked in it. Should I email the landlord and ask him about it, or do landlords get annoyed at that kind of thing? I'm not sure how to phrase the question either. (By the way, I have social anxiety disorder so I tend to worry a lot about these kinds of things.) What should I do?

(While at the same time being financially responsible so as to not end up bankrupt.)

Will she be called Queen Camilla? Because it sounds funny.

(I am from Canada. So I don't know how the monarchy works exactly. I know that she will be Queen Consort rather than Queen Regnant. But is it common for name changes to occur when heirs apparent ascend?)

If f is continuous on [0, pi], use the substitution [u = (pi - x)] to show that

the integral from 0 to pi of [(x)(f(sin x))dx]

is equal to

[pi/2] times the integral from 0 to pi of [f(sin x)dx].

It's really difficult to type these integrals onto Yahoo Answers, so it might help if you rewrite the question on paper. I've been trying to figure it out for some time now, and any hints would be wonderful.

Thanks.

(James Stewart's "Calculus" 6E, Section 5.5, Question 86)

I've read that brain cells can divide under certain conditions, and there are parts of the brain where cell division is a common occurrence. But I'm wondering whether or not the actual atoms that make up the brain remain over time. For example, will the atoms that made up my brain as a baby still be there when I die? Or will different atoms replace them? How much does the human brain change over time anyway?

I'm trying to look at this from a philosophical standpoint. For example, if my personal identity is purely based upon a certain configuration of the atoms making up my brain, and if those atoms are replaced by new atoms over time, then one could argue that I am a different person because I don't have the same brain that I did at an earlier time.