TA Bong_Lee>>the double pro.Yan Min seems be fixed!
Prof Yan--Min-->>Hello! Nobody is talking?
TA Bong_Lee>>in fact on last time
TA Bong_Lee>>seldom people talk
Prof Yan--Min-->>How are you doing? Derek?
Chun Pong--Derek--Chow>>i am reading the course materials
Prof Yan--Min-->>Which part?
Chun Pong--Derek--Chow>>linear transformation
Prof Yan--Min-->>You just started reading these?
Prof Yan--Min-->>Hello, SH Lau!
Chun Pong--Derek--Chow>>i am busy with my school work, very exhausting, so i am sorry that i read it so late...
Prof Yan--Min-->>Is the linear algebra material too much?
TA Bong_Lee>>Maybe we need to slow down!
Prof Yan--Min-->>I will ask the students on Saturday about the progress.
Siu Him--Lau>>i think the material for today is too much
Siu Him--Lau>>
Prof Yan--Min-->>what about the material for Monday?
Chun Pong--Derek--Chow>>i think i want to have more practice
Prof Yan--Min-->>Isn't the exercises enough?
Chun Pong--Derek--Chow>>enough
Prof Yan--Min-->>Then what do you mean by more practice?
Chun Pong--Derek--Chow>>but i just think the problem in mind, but not on paper...... it's just my problem~
Siu Him--Lau>>me too
Prof Yan--Min-->>It is indeed very important to practice by yourself.
Prof Yan--Min-->>I saw we just had two new students joining in. Hello!
Chun Pong--Derek--Chow>>why is that only Holap student come here?
Chun Pong--Derek--Chow>>only 4...
Prof Yan--Min-->>Where is Holap?
Chun Pong--Derek--Chow>>it's our school.... in San Po Kong
Prof Yan--Min-->>Do you know each other?
Chun Pong--Derek--Chow>>we are in the same class, luckily
Siu Him--Lau>>all in same class
Prof Yan--Min-->>Well, do you study together?
Chun Pong--Derek--Chow>>only me study chemistry, other study applied maths
Prof Yan--Min-->>It is easy to get lazy in a self-paced course. Is it possible for you four to set up a specific time, like one hour each day, to study linear algebra together?
Prof Yan--Min-->>It is always more efficient to study together.
Siu Him--Lau>>one hour each day may be too much
Chun Pong--Derek--Chow>>actually, we are still not able to cope with our pure maths.....not even linear algebra
Prof Yan--Min-->>Do you think my lecture notesdifficult to understand?
Chun Pong--Derek--Chow>>no
Prof Yan--Min-->>Any opinion from the other students?
Siu Him--Lau>>it is clear but i still need to read them many times to understand them
Prof Yan--Min-->>Do the other two students think the material difficult?
Chun Pong--Derek--Chow>>silent...
Prof Yan--Min-->>Derek, Are you buddys less ourspoken than you?
Shuk Fan--Chau>> quite difficult
Prof Yan--Min-->>Can Shuk Fan be more specific?
Siu Him--Lau>>some part is similar to what we have learned in pure maths
Chun Pong--Derek--Chow>>onto and one-to-one
Shuk Fan--Chau>>self study is difficult, since i am quite lazy
Prof Yan--Min-->>Derek, do you mean you have difficulty understanding onto and one-to-one?
Prof Yan--Min-->>Shuk Fan, self-study is indeed not so easy, especially for a conceptual math course. That is why I suggest you four to get together regularly.
Chun Pong--Derek--Chow>>actually i am studying linear transformation. i have just glanced at the topics
Prof Yan--Min-->>Derek, you said the material is not difficult. right?
Chun Pong--Derek--Chow>>not today's and some parts on monday, and i need real practise to ensure that i know it well
Prof Yan--Min-->>From the talks so far, it appears to me that the difficulty is not the material itself, but the time and self discipline. Am I right?
Siu Him--Lau>>in fact i cant follow the steps of my teacher in school. so i dont have much time to study here
Chun Pong--Derek--Chow>>totally true
Prof Yan--Min-->>Is the high school stuff very difficult?
Prof Yan--Min-->>What subject you cannot follow?
Shuk Fan--Chau>>pure math
Siu Him--Lau>>pure maths is most difficult
Prof Yan--Min-->>Hmm. Why not?
Chun Pong--Derek--Chow>>i can't follow my maths teacher's maths language
Ka Wai--Chan>> Yes
Prof Yan--Min-->>Do you think you have a bad teacher?
Shuk Fan--Chau>>no
Chun Pong--Derek--Chow>>no
Ka Wai--Chan>>NO
Prof Yan--Min-->>Then what is the "language problem"?
Siu Him--Lau>>i need to ask him after class in order to understand what is injection, surjection and bijection.
Prof Yan--Min-->>Oh. You are learning injection, surjeciton, etc?
Chun Pong--Derek--Chow>>i believe my maths teacher comes from planet MATHS
Shuk Fan--Chau>>why?
Siu Him--Lau>>is that they are similar to transformation?
Prof Yan--Min-->>What do you mean? Derek?
Chun Pong--Derek--Chow>>sometimes we can't communicate
Prof Yan--Min-->>Here is a short but useless answer: Injection = one-to-one. Surjection = onto.
Siu Him--Lau>>he is going too fast
Prof Yan--Min-->>If you have difficulty understand injeciton and surjection, perhaps it is good time to read my lecture notes on onto and one-to-one.
Prof Yan--Min-->>Do you have opportunity to give feedback (such as "you are going too fast") in the clkassroom?
Siu Him--Lau>>Your notes are easier to understand because there are more example.
Prof Yan--Min-->>Thank you for you good words on my notes. I offered more example in order to help you study by yourself.
Ka Wai--Chan>>Yes the examples are really good with links
Prof Yan--Min-->>Maybe you could tell your teacher my lecture notes. He may use my examples in the class.
Prof Yan--Min-->>I asked about giving feedback to your teacher in class. Do you have any answer?
Siu Him--Lau>>i ask him after class but not during lesson.
Prof Yan--Min-->>Did you tell him he was too fast?
Chun Pong--Derek--Chow>>i think he gives so few examples about tackling the questions
Ka Wai--Chan>>I think the situation is that he teaches us the concept and we work harder at home
Prof Yan--Min-->>I see. It appears that you learn math only in theory, but not much practice and examples.
Siu Him--Lau>>i think asking face-to-face is more useful.
Siu Him--Lau>>then i know exactly what my problems are.
Prof Yan--Min-->>But I think starting from concept, with few concrete examples, is very difficult.
Prof Yan--Min-->>It does not seem to be a face-to-face problem. You did not see me. But you can learn something from my lecture notes.
Prof Yan--Min-->>Still, I think the right way to understand concepts is more important.
Chun Pong--Derek--Chow>>and more practice is important
Prof Yan--Min-->>Yes you are absolutely right. First learn concepts based on concrete examples. After you feel you understand them, practice with concrete problems.
Prof Yan--Min-->>Theories and concepts are not coming from nowhere. They are the abstractions of concrete, common sense examples. So the right way of learning them is through examples.
Prof Yan--Min-->>I noticed a new participant. Hello!
Prof Yan--Min-->>Where are you from? Hiu Bga?
Prof Yan--Min-->>Sorry. Hiu Nga?
Prof Yan--Min-->>Any math problems?
Prof Yan--Min-->>Any problems from your high school class?
Siu Him--Lau>>yes but it is related to probability
Prof Yan--Min-->>Would you please tell me?
Siu Him--Lau>>wait a minute
Prof Yan--Min-->>No problems from the others?
Siu Him--Lau>>4 numbers are chosen at random from the first 2n natural numbers. Find the chance that the sum of two of the numbers thus chosen is equal to the sum of the other two.
Prof Yan--Min-->>Do you mean the sum of the first two is the sum of the last two?
Chun Pong--Derek--Chow>>what a difficult problem!
Shuk Fan--Chau>>yes! it's one of the question in the homework, very difficult
Prof Yan--Min-->>It does sound difficult. It will take me a whole to think through. I have to confess I am not good at probability.
Prof Yan--Min-->>Well. I should say complicated, but not difficult.
Chun Pong--Derek--Chow>>luckily i study chemistry
Siu Him--Lau>>i nearly court all the cases
Siu Him--Lau>>count
Prof Yan--Min-->>OK. Let me try. Let the four numbers be a, b, c, d in the ascending order. Then d = b + c -a. So the problem is really the random choices of a, b, c satisfying b + c - a no bigger than 2n.
Prof Yan--Min-->>Fixing a first. How many b and c do we have?
Prof Yan--Min-->>It is the number of (b, c) satisfying 1 <= b <= 2n, 1 <= c <= 2n, b + c <= a + 2n, where <= is less than or equal.
Prof Yan--Min-->>I forgot to add b <= c. You should know how to get this number (draw the reagion where (b, c) is on the plane. I leave you to find this number, call this number N(a).
Prof Yan--Min-->>Then the totoal number of choices is N(1) + N(2) + ... N(2n). Divided by (2n)^4, you get the probability.
Prof Yan--Min-->>Sorry, I also fogot to add the condition a <= b in counting N(a).
Prof Yan--Min-->>The method I outlined may not be the most clever one. But it should work.
Hiu Nga--Cheung>>how to prove' (n)(n+1)(n+2)(n+3) is divisible by 4!' for n>=1
Hiu Nga--Cheung>>by MI
Prof Yan--Min-->>What is MI?
Chun Pong--Derek--Chow>>mathematical induction
Prof Yan--Min-->>Again your method may not be the most clever one. But it indeed works.
TA Bong_Lee>>there r 2 even number!
Hiu Nga--Cheung>>i did it wrongly
TA Bong_Lee>>it's the hint
Prof Yan--Min-->>The tricky way is to realize that n(n+1)(n+2)(n+3)/1 2 3 4 is the number of ways of choosing four ball from n+3 balls. The number of choices is an integer.
Hiu Nga--Cheung>>what do you mean
Prof Yan--Min-->>The theory of choosing balls should be the first thing you learn in probability.
TA Bong_Lee>>n+3C4 is integer !
TA Bong_Lee>>combination!
Siu Him--Lau>>very fast method
Prof Yan--Min-->>Oh. Perhaps you are confused by my use of 1 2 3 4. I mean the product of the four, which is 24.
Siu Him--Lau>>4!
TA Bong_Lee>>n+3C4=n(n+1)(n+2)(n+3)/1 2 3 4 =(n+3)!/[4!(n-1)!]
Chun Pong--Derek--Chow>>really fast method
Siu Him--Lau>>but the Q ask him to use MI.
Chun Pong--Derek--Chow>>but how about MI?
Hiu Nga--Cheung>>it is her not him\
Siu Him--Lau>>i am sorry
Siu Him--Lau>>i dont understand why we need to use MI.
Hiu Nga--Cheung>>homework
Chun Pong--Derek--Chow>>just like our homework EX 3.1 Q1
Prof Yan--Min-->>Certainly MI is the first natural thing one should try. I am not sure how easy it is for this problem, though.
Chun Pong--Derek--Chow>>if we don't use MI, just 3 STEPS
Prof Yan--Min-->>What steps?
Chun Pong--Derek--Chow>>sorry, i am just talking about my homework tonight
Prof Yan--Min-->>Does anyone need my help?
Siu Him--Lau>>one more student here
Hiu Nga--Cheung>>yes
Prof Yan--Min-->>I just realized MI on n(n+1)(n+2)(n+3) divisible by 4! is somewhat complicated.
Prof Yan--Min-->>In fact, what one needs is a double MI on the following problem: n(n+1)...(n+m-1) is divisible by m!
Chun Pong--Derek--Chow>>OH what's that?
Prof Yan--Min-->>First it is easy to see the statement for m=1: n is divisible by 1.
Prof Yan--Min-->>Now assume the m-th statement is true: n(n+1)...(n+m-1) is always divisible by m!. Let us try to prove n(n+1)...(n+m-1)(n+m) is divisible by (m+1)!
Prof Yan--Min-->>Note that my previous line is an inductive step already
Prof Yan--Min-->>In order to rpove this one tep, we induct on n again (note that m is fixed in this step.
Prof Yan--Min-->>The key becomes: Assume n(n+1)...(n+m-1) ia always divisible by m!, Assume n(n+1)...(n+m-1)(n+m) is divisible by (m+1)!. Under both (industive) assumptions, prove that (n+1)(n+2)...(n+m)(n+m+1) is divisible by (m+1)!
Prof Yan--Min-->>Boy, they really should not ask you to do this exercise.
Chun Pong--Derek--Chow>>i think it is out syllabus, right?
Prof Yan--Min-->>Yes and no. Double MI is to put two MI together. So it is an extension of the usual MI. However, the teacher should have done a similar example before asking you to do this.
Prof Yan--Min-->>The high school math looks quite tough to me.
Siu Him--Lau>>i am sorry. we are going to prove (n+1)(n+2)...(n+m)(n+m+1) is divisible by (m+1)! or (m+2)! ?
Prof Yan--Min-->>Note that m is already fixed in the previous inductive step. So we keep (m+1)! and not jumping to (m+2)!
Hiu Nga--Cheung>>f.6 student should do it in a simpler way
Prof Yan--Min-->>Actually the (single) MI on original 4! problem should lead to the 3! (dividing n(n+1)(n+2)) problem. Then to prove the 3! problem, we use induction again. This reduced to 2! problem. I think your teacher expects you to do that.
Hiu Nga--Cheung>>that means 5! lead to 4!
Prof Yan--Min-->>You got it. 6! reduce to 5!, 5! then reduce to 4!, etc. This is one level of induction (the induciton on m above). Then each reduction from (m+1)! to m! requires another induction. Hence dhouble MI
Prof Yan--Min-->>Any of you finished reading on-to-one?
Prof Yan--Min-->>Any of you enrolled in similar web based course before?
Chun Pong--Derek--Chow>>this is new to me
Siu Him--Lau>>i haven't.
Prof Yan--Min-->>Why didn't you choose the other courses (computer, physics)?
Chun Pong--Derek--Chow>>i think astro phy is much more difficult
Prof Yan--Min-->>Math may not be the most difficult. But it definitely requires the most effort.
Siu Him--Lau>>if i attend more courses, i will be reading notes and books all over night
Hiu Nga--Cheung>>i am leaving
Prof Yan--Min-->>If there is no more message. I might be leaving.
Chun Pong--Derek--Chow>>then see you on saturday morning~
Siu Him--Lau>>Good bye
Prof Yan--Min-->>Bye
Chun Pong--Derek--Chow>>gtg, goodnight everyone~
Shuk Fan--Chau>>good night everybody
TA Bong_Lee>>bye