"Do you prove all theorems whilst studying? When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my studies -- sequence two of Calculus -- but what I'm trying to understand right now, though, is how one actually goes about studying so that when finished with a good text, there's more of an intuitive understanding than superficial. After reading ""The Art Of Problem Solving"" from the Final Perspectives section of part eight in 'The Princeton Companion to Mathematics', it seems to hint at approaching studying in that very way. A quote in particular, from Eisenstein, that caught my attention was the following -- I'm not going to paraphrase much: I feel

atarentspe 2022-09-11 Answered
Do you prove all theorems whilst studying?
When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my studies -- sequence two of Calculus -- but what I'm trying to understand right now, though, is how one actually goes about studying so that when finished with a good text, there's more of an intuitive understanding than superficial.
After reading "The Art Of Problem Solving" from the Final Perspectives section of part eight in 'The Princeton Companion to Mathematics', it seems to hint at approaching studying in that very way. A quote in particular, from Eisenstein, that caught my attention was the following -- I'm not going to paraphrase much:
The approach utilized by the director turned into as follows: each pupil needed to show the theorems consecutively. No lecture took place at all. nobody became allowed to inform his answers to every person else and every scholar acquired the following theorem to show, impartial of the opposite college students, as soon as he had proved the previous one correctly, and as long as he had understood the reasoning. This changed into a completely new interest for me, and one that I grasped with outstanding enthusiasm and a zeal for knowledge. Already, with the first theorem, i was a long way ahead of the others, and whilst my friends had been still struggling with the eleventh or 12th, I had already proved the hundredth. there has been simplest one younger fellow, now a medicine pupil, who could come close to me. even as this technique is excellent, strengthening, as it does, the powers of deduction and inspiring self sustaining wondering and opposition among college students, generally speakme, it can likely no longer be adapted. For as lots as i can see its advantages, one ought to admit that it isolates a positive power, and one does not obtain an overview of the complete concern, that may simplest be done with the aid of an amazing lecture. as soon as one has acquired a amazing variety of material thru [...] for college kids, this technique is manageable only if it deals with small fields of effortlessly, comprehensible information, in particular geometric theorems, which do now not require new insights and thoughts.
I feel that this type of environment is something you don't often see, especially in the US -- perhaps that's why so many of our greats are foreign born. As I understand it, he does go on to say that he wouldn't particularly recommend that method of study for higher mathematics, though.
A similar question was posed to mathoverflow where Tim Gowers (Fields Medal) went on to say that he recommended similar methods to study: link
I'm not quite certain that I understood the context of it all, though. Upon asking a few people whose opinion mattered to me, I was told that it if time were precious to me, it would be a waste going about studying mathematics in that way, so I'd like to get some perspective from you math.stackexchange. How do you go about studying your texts?
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Answers (1)

coccusk7
Answered 2022-09-12 Author has 14 answers
There is a continuum in the way one understands a theorem.
At one end of the spectrum mathematicians just try to understand the statement and use it as a black box .
At the other end they understand the theorem so well that they improve on it: this is called research.
An important thing to keep in mind is that your attitude toward a result is not fixed for ever: you may first consider it as a black box and solve exercises by blindly using it, then see how it is quoted in proving corollaries or other theorems and finally come back to it and realize that it is actually quite natural.
Professors have the advantage that they really have to understand a theorem if they want to teach it well and answer the students' questions.
One of the great aspects of this site is that everybody can be a teacher: I strongly advise you to try and answer questions here. They are at all possible levels and I am sure you can find some that you will answer very competently.
A paradoxical way of expressing what it means to have understood a theorem is to say that ideally you have to reach the stage where you consider that all its proofs in the literature are "wrong": it is a patently absurd statement but it conveys the idea that the theorem is now yours because you have integrated it into your own mathematical world.
Edit
Since Neal asks about this in his comment, let me emphasize that when I say that proofs in the literature are "wrong" I mean that, although they are technically 100% correct, they don't correspond to the subjective way one has organized one's understanding of the subject.
For example, the definition I like for a finite field extension K/k to be separable is that it is étale i.e. that the tensor product with an algebraic closure of k is split: K k k ¯ k ¯ n
I know this is rather idiosyncratic and of course I know the equivalence with the usual definition, but then I feel that long proofs that C R C is not a field are "wrong" since I know, by the definition of separable I have interiorized, that C R C = C 2

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New questions

Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
x ( k + 1 ) = A x ( k ) + b u ( k ) , y ( k ) = c x ( k )
where b = ( 0 , 1 ) T , c = ( 1 , 0 ) , A = [ 2 1 0 g ] for some g R
Find a feedback regulation (if there is any) of the form u ( k ) = K x ^ ( k ) where x ^ ( k ) is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders e ( k ) = x ( k ) x ^ ( k ) go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are λ 1 = 2 , λ 2 = g (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle | z | = 1.
The state observer equation is
[ x ( k + 1 ) e ( k + 1 ) ] T = [ A b K B k O A L C ] [ x ( k ) e ( k ) ] T
With characteristic equation
χ ( z ) = | z I A + b K | | z I A + L C | = χ K ( z ) χ L ( z )
Also consider
K = [ k 1 k 2 k 3 k 4 ]
and let a = k 1 + k 3 , β = k 2 + k 4
Then χ K ( z ) = ( z 2 ) ( z + g + β ) + a.
So we can select some eigenvalues inside the unit circle and determine a , β in terms of g. Choosing e.g. λ 1 , 2 = ± 1 / 2 we get a = 3 g + 33 / 8 , β = 9 / 4 g , g R
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose | λ 1 , 2 | < 1?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)