Ready for some linear algebraic trivia? Let B be a bilinear form on
R^n. An nxn matrix g is called an
automorphism of B is
B(gX, gY) = B(X, Y) for all X, Y in
R^n. Show that, for any bilinear form on
R^n, the collection of automorphisms (with
the regular matrix multiplication) forms a group. This is called the
orthogonal group of the form B.
Hint: Let G be the above collection. One way of proving this is to show the following three things: the identity matrix is in G, if g is in G, then g^-1 is in G, and if g,h are in G, then so is the product gh.
It should be fairly straightfoward. Good luck!
I'm cooking terriayki salmon steaks and chocoloate devil's foodcake for dessert. I also picked up a spinach salad to start off with. Mmmmm... happy friday everyone!
I hope you got the right answer in yesterday's blog. I might have another one for you later.
In any event, I did go to the dentist's office today. No cavities! That calls for lots of ice cream (:
I was heating up some beef stew in my microwave this evening and it exploded. Well, a piece of beef exploded. Right through the plastic wrap. Oh well. The stew was still good, though.
I learned a new magic trick today. If I were to perform it for you, this is what you would see:
I would take a red container in my left hand, and place (one by one) 3 quarters in it.
I would then close my hand and shake it near your right ear (assuming you were facing me).
You would then hear the coins jingling (unless you were deaf).
I would then place the red container with the coins in my right hand.
Once again, I would close my hand it shake it near your left ear (*jingle, jingle*)
Then, placing my left hand beneath my right hand, I would ask you to tap my right hand with your finger.
You would so, I hope.
Then, when I reveal both hands, you would see the three coins in my
left hand, and the empty container in
my right hand.
Aren't you impressed?
Computation proofs are the best. Given a set A with an commutitive multiplication, define [a, b] = ab - ba.
Now prove that any three elements of A satisfy the Jacobi Identity:
[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0.
It's fun, trust me.
Matthew Thomas, author of
Before and After has agreed to be an official Friend of FlyingSheep! This means that he will have the option of answering your questions along with my other friends.
Wow, an author, a cartoonist (Fluble), and all sorts of artistic and creative people. This should be fun.
My dentist's appointment was really for tomorrow. oh well.
Had my first class in Lie Algebras today. It was good. Mostly a review of Lie Groups and some basics with tangent bundles, and that [X, Y] multiplication. We also defined e^X (For an nxn matrix X) and derived the usual formulae (in particular det(e^X) = e^trace(X). There are about 8 of us in the class, which is relatively huge. At some point I'll talk about the anti-social traits of mathematicians. Not anti-social in a bad way, at all. There's no ill will behind it, which makes it all the more interesting and less obvious.
Also, I thought of making an inventor's little helper for FlyingSheep. It would step you through some thought exercises geared towards helping you invent the next big thing. It's only a germ of an idea right now, and it'll have to incubate considerably before it's ready to be implemented.
Anyways, I've got a dentist's appointment soon, so I've got to go. Wish me luck!
Collection of thoughts about games on my shelf:
- Diablo rocks. I have not yet begun Diablo II, but I look forward to it anxiously. The Jedi Princess and I are playing through Diablo together (we're now at about level 25, just entered Hell), and it's a lot of fun.
- Nox is a great game. I've finished it as a Conjuror (cheating a bit through the last level), and I'm now about 2/3 of the way through as a Warrior (which is not as interesting by far). The Jedi Princess plays this game as well, but we don't have the computer power (yet) to play multiplayer.
- Enemy Infestation is another Diablo knock-off. I haven't really been able to get heavily involved in this game yet.
- Warcraft II is one of the classics. The single-player mode has little attraction now, but I still enjoy playing it with other people.
- Gearheads is a game I played once and put away. That is, until my brother came to visit a few weeks ago. He and I played it and somehow he got me hooked on it again. I particularly like the combination of the bulldozer and the wind-up hand.
- I'm going to finish Half-Life someday, I know it. I didn't get very far into it when it simply became too frustrating for me.
- Dungeon Keeper is a great game, but I suffer from some strange bug - Every so often, my mouse will pick a direction, and get stuck traveling in that direction. At this point, both the keyboard and the mouse are completely unresponsive. I just have to wait (or reboot) until the computer decides to let me play again. You can't imagine how frustrating it is to sit by helplessly while your dungeon (which you haven't saved in ages) gets destroyed.
- I suppose you won't be surprised to learn that I didn't get very far in QIII, either. I think I was too distracted by Starcraft at the time.
- Sin is another one that I'll have to get back to when I learn how to play FPS's.
- On the other hand, I've been playing a lot of the classic FPS's, especially Doom and Heretic. They rock.
- I'd be remiss if I didn't mention Sheep. It's cute, it's fun, and it's a puzzle game, which I like, but the controls are sometimes too frustrating. I'll play a level a few times to get it right, but I rarely play more than one level at a sitting.
- Finally, from left field, a game that you've probably never heard of - Pinky and the Brain's World Conquest. The best part of this game, undoubtedly, is the clips that you see upon completing every other level or so. It's better than the pellets that they give rats after they run their little races.
From James Adams'
Conceptual Blockbusting:
"Why should man be creative?"
"Because creativity allows self-actualization."
"What good is self-actualization?"
"It allows a man to be happy."
"What is happiness?"
"Well-being."
"What is well-being?"
"Go to hell."
The Star Wars Chair came out great, of course. It has a Princess Leia on the seat of the chair. There are three horizontal strips of wood forming the back of the chair. The middle rung has the Death Star on it (with various starfighters in battle around it), and the bottom rung has Alderaan on it (Leia's home planet). The top rung (not yet painted) will have the rebel insignia on it, I think.
I've got two other similar chairs here (and I think there's at least one more in storage) that will be painted eventually. I'll keep you posted. I'm hoping to get a FlyingSheep-themed chair, but I'm tempted to just have one chair for each of the original three movies.
I just ordered a larger monitor for my home use. I'm taking a leave from work to finish up my master's degree in mathematics, so I'm going to be doing more computer stuff from home (especially with respect to FlyingSheep.com). That's my justification. Of course, it won't hurt the game-playing experience, but that's
not my primary concern. Heh, heh, heh.
While I'm writing this, the Jedi Princess is painting one of my chairs to have some star wars figures on it. She's working on Princess Leia right now. It looks really good. I'm hoping that she'll do a FlyingSheep-themed chair as well at some point. I can't wait until it's done so I can look at it.
G. Polya wrote the following in
How to Solve It:
Rules of Style:
The first rule of style is to have something to say.
The second rule of style is to control yourself when, by chance, you have two things to say;
say first one, then the other, not both at the same time.
