How do I close the gap between intuitively knowing something is true vs being able to prove it?

Question

For example, one of my review problems is: Let $S_k$ be the kernel of $T^k$. Show there is a $K$ such that $S_K = S_{K+1} = \cdots$

Somewhere in the back of my brain there's an intuition that told me, "Well duh, $K = \dim(T)$." Obviously (to me anyway) once $K$ gets bigger than $\dim(T)$, you're either cycling around in some $T$-invariant subspace, or you're in the null space. My brain got there by envisioning a matrix in my head and thinking about what would happen to each vector in the domain until it was satisfied that $K=\dim(T)$ satisfies the prompt. At this point that part of my brain was content that a solution exists and moved on to something more interesting. But if I sit down and try to prove it with only the properties of transformations and subspaces, I don't even know where to start. Do I take a basis? Do I count dimensions? Nothing I try seems to get me anywhere.

I think my brain is thinking about it the wrong way. Lower division math is about manipulating formulas and calculating. I got pretty good at doing that, and now my brain seems to attack every problem that way. I get the sense from talking to other (smarter) people that proofs are different. When my instructors come up with proofs they seem to be doing something completely different in their minds than I'm doing. To me it seems more akin to solving a puzzle than to manipulating equations. I don't see what they're doing that makes it so clear to them, in the way that lower division stuff is clear to me.

I've heard many tips, including "Write the first line and the last line of your proof, and then try to fill in the gaps." And also, "Write statements for everything you know is true in one place." And also, "Write as many statements as you can until you see something that can help you make the conclusion you need." And so on. Those are good tips that help simplify the problem, but I feel like the real solution is rewiring my brain to think in a different way. Sitting here and looking at and doing dozens of proofs hasn't gotten me anywhere, so I'm hoping for some insight from some people smarter than I am.

Answer 1

Your professors are not thinking much differently than you can. But the proofs they are supplying are (most of the time) the results of somebody thinking about the problem "intuitively", seeing why the proposition "has to" be true, then putting each step in that intuition into a justifiable statement. That last step is the one you are having trouble with.

So there are two tough issues. The first is to be able to express steps in your intuitive reasoning as steps in a proof. For example, when you think "$J(K)$ is cycling around in some subspace" you have to realize that means that if you have the set of previous values of $J(n)$, then $J(K)$ can be expressed as a linear combination of the previous values. So at some point you be saying something like "Let $W(n)$ be the space spanned by ..." and then, since your intuition is guiding you correctly, you will have to say that some new vector(s) must be expressible as a combination of the basis vectors of $W(n)$ if the dimension is at least $K$.

The second tough issue is that in a proof, you have to fill in the boundary cases and making sure each step is airtight. And if you are doing real math, and not just a homework or test problem, there will be some times where in filling in these boundary cases, you will need to add restrictions to (weaken) your original proposition.

The last step in a really nice proof is to see how you can change your reasoning so as to make the proof clearer, shorter, or more elegant. That last part is truly an art, and may be a place where you don't have as much talent as some better mathematician.

Answer 2

We can prove the kernel of $T^n$ is a subspace of the kernel of $T^{n+1}$.

Proof: let $v$ be in the kernel of $T^n$, then $T^{n+1}v=T(T^n(v))=T(0)=0$ so $v$ is also in the kernel of $T^{n+1}$. since the dimension of the kernel is limited there must b e a point where the dimension of the kernel of $T^k$ is the same as the dimension of $T^{k+1}$.Since a finite dimension vector space cannot have a subspace with the same dimension as itself other than itself we have the kernel of $T^k$ and $T^{k+1}$ is the same. from here on suppose there is an element $v$ so that $T^m(v)=0$ but $T^k(v)\neq0$. Let $m$ be so that $T^{m-1}(v)\neq0$. Then we have $T^{m-1-k}(v)$ is not in he kernel of $T^k(v)$, but then it is also not in the kernel of $T^{k+1}(v)$ and so $T^m=T^{k+1}(T^{m-1-k}(v))\neq 0$ a contradiction.

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As to the other question, I think you could try to write down the ideas in your mind, try to make them concrete. Sometimes you have to be acquainted with the notation and sometimes it is practice.

For example: the first time I learned group theory I has no idea how to prove the statement if two identities exist they are the same. i tried it for about an hour and had no idea how to do it. Now if I see a similar problem I know what kind of things I should write.