This about those magnetic toys buckyballs. My apologies, I don't really know any physics, so sorry if my terminology is confusing.
If one lines up two lines of buckyballs they can be lined up with the same orientation, (ie as they are in the cube when you get them) or oppositely (so that...
I loved doing the excercises in Baby Rudin. The material covered in the text itself is really quite minimal, but I found the excercises take that limited machinery as far as it can go
So I was talking with a friend about a problem and noticed the following construction arose naturally:
For a category C, let F(C) be the category where objects are objects in C, arrows are finite lists of arrows, identity is the list {id}, and composition is defined by tensoring as follows...
How might one prove that for any degree n polynomial p(x) with coefficients lying in a field k, there exists any nxn matrix with entries in k with characteristic polynomial p?
What definition are you using? Show that this extension is normal is really not so bad. (hint: start with any polynomial which has a root, and find an automorphism taking it to other roots...)
In this case the problem is indeed whether or not they are collinear, but more generally the problem is to figure out whether they are independent. As for finding the coordinates of V relative to that basis, what do coordinates mean? The coordinates are two numbers a and b such that
V=aV1+bV2...
Well, presumably he first noticed separately that e^ix and e^-ix work. He may have found this simply by trying things, but more likely, he realized that the equation says that y''=-y, which says that the second derivative is proportional to the original function. One obvious thing to try would...