Recent content by A David

I should note that Sarason is seriously lacking in exercises.

Of course.

I studied art my freshman year, then general studies at a community college. When I got into a university, I started at trig and worked my way up from there. Since I got a two year degree from a community college, my GPA started from scratch at the university so my previous grades didn't have a...

Yes, but that's a different map. I wouldn't say that it's not important, only note that ##\mathbf{q}_\theta \mathbf{v} \mathbf{q}_\theta^{-1} = \mathbf{q}_{-\theta}^{-1} \mathbf{v} \mathbf{q}_{-\theta},## so the inverse map takes us from left to right or vice versa. You can call it "left" or...

Notice that this is equal to ## \cos{\frac{-\theta}{2}} + u_x\sin{\frac{-\theta}{2}}\mathbf{i} + u_y\sin{\frac{-\theta}{2}}\mathbf{j} + u_z\sin{\frac{-\theta}{2}}\mathbf{k}##. It only depends on whether you want ##\theta## to correspond to a clockwise or anticlockwise rotation. If we have a...

Sorry, I was busy earlier and was only able to answer partially. It turns out that the quaternions are isomorphic not to numbers but to sets of matrices with complex entries. In fact, we know that all groups are isomorphic to a group of matrices with complex entries. The quaternions are...

Isomorphic is from the Greek, meaning "same form." Formally, it means that there exists a bijective map between the sets that preserves the operations of the group (for a group isomorphism) or ring (for a ring isomorphism). Informally, two objects that are isomorphic can be thought of as the...

We don't need an axis to rotate around in 2d, a point works just fine. 3d real space is special because we can equip it with the vector cross product. We can do an analogous operation in higher dimensions but we don't get a vector back. Notice that the quaternions behave just like taking...

Any mapping ever, really.

A lot of sign errors can be avoided by using parenthesis a lot when working with multiple terms.

After the first few years it will be very different and much more interesting. Learn how to program well and soon, and get involved with your professors as an undergraduate research assistant. Ask them what their research is like! Its what they do for a living. If you like physics, do...

I just noticed that your question was specifically about masters programs, which I have less experience applying to (...despite the fact that I'm in one).

Basically, Fourier showed that all periodic functions can be broken down into a sum of simple sinusoidal waves of various frequencies. (Think superposition principle, if you know it) The Fourier transform takes a function and gives you the coefficients of the terms in that sum. An...

You can think of the standard deviation as "the average distance from the average."

I have similar stats, including a publication, and no GRE subject. My undergrad was from a small and new program. Last year I applied to PhD programs at Duke, UT Austin, Vanderbilt, and Arizona. I got turned down outright at Duke, and wait-listed everywhere else, but didn't end up getting in...