For the past 3 weeks, I’ve been in exam period. Having already taken four of my five exams, one would think that I’d be nearly relieved of all the stress involved with exam period. However, it is not really so.

Let me visualize this: it’s like a steeplechase course. Steeplechase and not hurdles, because hurdles are all the same size. And this steeplechase course is one with five hurdles, one for each exam, but imagine the solar system’s first five planets, and their relative sizes. Now match these up, respective to each hurdle. This is how I went into this exam period: First exam, ah piece of cake. Second one.. not too bad. Third one, ok slightly harder. Fourth one, ah no problem.. and then I look ahead. In the distance, there looms the last hurdle, heavy and thick, oozing with malice, weapons of confusing gas, to make you completely lost in its midst, if by chance you can’t catapult yourself over the top.

This last hurdle is 2 days away: Algebra/Geometry exam – Wednesday, May 25th, 2011. But I am not completely despondent.
This was my strategy, one which I hope really has paid off. First, I have been keeping track of how many hours I study specifically for this exam per day. This numeric is in order to keep me motivated, and to have a goal to which I hold myself. Secondly, I recognized that my greatest difficulty is the fact that all the notes and course is written in french. In order to combat this, I have been diligently translating the important points in each chapter. First I tried to translate all of the text, but realized that dwelling on little sections that are not too important was less efficient than if I just went over the propositions, theorems, corollaries, and remarks written in bold. So after two chapters entirely translated, I rewrote just the bold text. When I stumbled upon notation that I didn’t get within an instant, I highlighted it separately, and tried to visualize, or in case it was just something different than what I was used to from the USA, I’d clarify in the translated notes.

To keep my brain from passing over information, I used other kind of tactics intermingled with this translation. These varied from copying demonstrations, then trying to rewrite them myself, to rewriting the correction of the midterm which I had failed miserably in March. In fact this last one was quite a confidence booster because as I rewrote the answers, they made complete sense, and really didn’t seem as convoluted as it seemed to me that day in March.

Now the killer hurdle doesn’t seem as impossible anymore. In any case, I will have done almost everything in my power to get the highest score I can.

Current tally of hours: 35.5
Goal before exam: 40

Update: total hrs: 52, exam score: 7.06/20. => need for oral exam with teacher.
Studied another 10 hrs before it the oral exam. Result:
Oral exam completed: Je l’avais déchiré. 🙂

My interpretation of Affine Space (2)

In section one, I introduced M (Manfred) and ū (ūlina), who are N’s (Norbert’s) parents, and Norbert and ānna are Paprika’s parents.. but if you ask P personally: “Hey P, who are your parents?” He will say, well of course N is my dad.. but he wouldn’t mention his mom, because his family tree looks like:

In this world, Males are the only family members (points), with females being the links between the males (vectors).

But one would ask, didn’t we just say that Mū can equal one and only one N… Yes. But aren’t there many points that can lie on a vector, perhaps lying on ū? i.e. a point Andy can be an element of ū?… yes

But isn’t this a contradiction? No.

Because … a woman COULD have many children in her “life” (or length of this vector – that assumes she dies after having her child… sad), aka she could have an infinite amount of eggs (according to newest research); thus Andy (A) is element of ū looks like: .

If Manfred and ū had mated earlier, then A would have been born instead of N, but as a consequence, the life of ūlina would have been shorter (visually, the vector ū would have been shorter by a factor of lambda λ. Let’s write this hypothetical situation mathematically:

Mλū = A.

so if lambda = 0.5, A would have been born only if M and ū mated at half the life-time of ū.

To come full circle, Now imagine the E, Ē and ExĒ (Affine space with afore mentioned properties) are permeable. Like layers in Adobe Photoshop, we can place one on top of the other. If the layers are not completely opaque, the layer below will show through.

The purpose of this is to point out that every time a point like M and a vector like ū mate and create N, N is in fact also in E. It’s like shining a light through E -> Ē -> ExĒ.

My interpretation of Affine Space

A little note: This interpretation uses a kind of metaphor that like most metaphors, may not work 100% of the time. When I define something in this world, just accept it. (i.e. if I say in this world, all humans have one arm, just go with it, and imagine the world with one armed humans as normal.)

Above I have drawn two worlds of random sizes, one of them is made up a ton of points, and the other of a bunch of vectors (directions). The + and • next to Ē means that these vectors can be added and multiplied without losing the characteristics of Ē. Think of Characteristics as a genetic trait… so if two humans mate, they have the same genetic “properties” – saying that their kid (in a perfect world we’re used to), just like the two of them, will have only two eyes, one mouth, and one nose (yes they may be different sizes and shapes, but that doesn’t change that there are two eyes for example).

Going back to E and Ē. These are “spaces” in mathematics, but we will equate them with two worlds. Each point lives in his little house on E, and each vector lives her life peacefully in Ē.

One day, a point in E we’ll call Manfred, and a vector in Ē called ūlina meet. Instantly they fall in love, and get married. But in order to do so, they have to enter a car (in math known as an application), called psi Ψ, which takes two elements from the worlds of E and Ē and creates a new one called “Affine Space”.. so romantic, I know.. but it has some sense; after all, the word “Affine” comes form the Latin word affinis which means “connected with”[1].

This Affine plane (another space/world) contains both Manfred and ūlina, and every other pair of point-vectors that have decided to marry. This new and wonderful world makes it possible for Manfred and ūlina to mate. Us mathematicians are quite lazy, so instead of writing out: “Manfred and ūlina have mated”, we write Mū. M stands for Manfred, ū stands for ūlina, and that odd plus sign means “they have mated”.

Naturally, we’d like to know what the heck a child of a vector and point will be. Well in an Affine space, Manfred and ūlina’s child will only have the characteristics of Manfred – in other words, he will be a point. Conveniently, the result of Mū = Norbert.

Ah but this means that a child will never be female, and never be like ūlina. And sadly in this world, that’s the way life is.

Going back to the Ψ – car. Before they can get out of this car, the point Manfred and vector ūlina will have to have mated, and given birth to Norbert.

I note that N (Norbert) can only have M (Manfred) and ū (ūlina) as parents… that is, biological parents. In addition, in this world it is quite sad, but M and ū can only have one kid N. (the car Psi (Ψ) is a vehicle of BIJECTION: a one-to-one relationship)

Also, sometimes, like humans do, a point M falls in love with 2 vectors, ūlina and ānna… so what happens when he tries marrying both?

Well, this can’t be possible in the world of Affine Spaces, it’s like trying to inhale and exhale at the same time. So, when M and ū + ā(ānne) hop into the Ψ-car, since there are more than one females, there has to come out more than one child.

Us humans would say, well then duh, Manfred is a lucky man, he would just mate with ūlina and then impregnate ānna, wait till they pop out Norbert and Paprika. WRONG. Remember when I said Manfred and ūlina can have only one child? Well, that still is true, so how else can two kids come out of the Psi-car?





Hint: Norbert ānne =..?

in fact, time is not an issue in this world. ānne can wait her turn to mate with Manfred and ūlina’s child Norbert once he is born, then wait till Norbert and ānna’s child Paprika is born before they all step out onto the affine plane. So in mathematical symbols, this looks like so:

Ψ(M, ū + ā) = Ψ(Ψ(M, ū), ā) where Ψ(M, ū)=N, so

Ψ(Ψ(M, ū), ā) is the same as Ψ(N, ā), which is a clear case of N and ā mating, so out comes a kid I have above called Paprika, or P for short 🙂

Above I have outlined two properties (characteristics) of an Affine space that must always hold true for an affine space to be an affine space:

1. When A point and vector mate, they can only produce one and only one child. Conversely, a child (or point) can only have one vector as mother and one point as father, and no other ones.
2. When M falls in love with 2 vectors, the application of M and the two vectors is the same as the application of the application of one vector then the other which ultimately results in a second child P… this will become clearer in the next section.