How to prove using higher abstractions instead of diving into axioms or a little bit deeper?

Question

Sorry for the bad formulated question, feel free to edit it. I will explain my question here.

I try to reflect on my abilities of proving theorems to become better at this. That is why after reading almost all proofs I wonder that I would never think about it, instead I prove until I reach the core of contradiction itself. While people deduce a little bit and return with contradiction, I take another branch of deducing and go into smaller details.

For example, I was asked to prove if A, B and C are matrices, A is invertible and AB = AC, then B = C. What my professor did is just he multiplied the equation from the left side to the inverse of A, he got identity matrices, IB = IC, so B = C. But instead of doing so, I start to think what it means to be singular (has an inverse). It means there is no zero row, for example, then it means another thing, until I show that it is impossible for B and C to be different. I hope you got the idea. What should I do to avoid such details and use what I already have? It seems like I don't see higher levels things, like I don't see the forest for the trees.

If it necessary, I can show also another example of a proof with comparisons, but it is more messy and related to vector spaces.

Answer 1

Most dismissive answer: You're probably just still in unfamiliar territory. You don't have enough experience with linear algebra yet to see how all the pieces fit together. Do as user4894 said, keep looking at the problem from different angles; understanding and proof economy will come in time.

If you want a higher-level answer, you should try turning the car on before you open the hood to see if it works. In your example you're given that A is invertible. Stop right there. Ask yourself "What is the definition of invertible?" see if the definition provides a way forward. In this case, that's essentially the whole proof. For many exercises like these the articulation of the problem strongly hints at a clear path forward.

Alternative answer: A more useful question might be "How do you form proofs requiring fewer steps?" I offer this as an alternative, because in an axiomatic system it is extremely common to have statements mutually depend on each other. What is seen as a "higher level of abstraction" can often be made arbitrary through axiom selection. Good examples of this phenomenon are the many equivalent statements of the axiom of choice or closer to home the many equivalent statements of invertibility. So one method of making shorter proofs is to choose a stronger starting position.

One confounding element here is that what constitutes a sufficiently rigorous argument is often subjective. (e.g. professors omitting deductive steps that are deemed trivial.) So another way of making shorter arguments is to assume more on the part of the reader.

Hope this helps some! Crafting elegant arguments is an art.

Answer 2

Your professor is using the properties of the equality. If A=B then whatever we do to A it must equal the same thing done to B since we have said that A=B.

So if AB=AC then we can multiply by A^(-1) on both sides on the left. So A^(-1).AB=A^(-1).AC But now A^(-1).AB=(A^(-1).A).B=I.B=B And similarly for the RHS, A^(-1).AC=C So A=C

This proof is in essence no different from seeing if xy=xz then y=z in algebra and it this proof that informs the one above and you're expected to see and make use of this analogy. Mathematics is constructive in the way it builds upon previous achievements.

Its often the case we don't see the forest for the trees, especially in a subject like mathematics when the objects that we are concerned with are objects that are presented to the inner sense of the mind rather than the outer sense of actually seeing trees and forests. There the eyes come to our rescue showing us immediately that we have both trees and forests. For objects of the mind the mind has no such helper and must struggle alone. This is one reason why mathematics is such a difficult discipline.

Poincare once told of a story when he struggled with proving a property of Fuschian functions. He sat down every morning to work with them and when the afternoon came he would give up. This went on for several weeks and then one day all these came together at once and he could see where the problem lay and solved the problem more or less at once.

So the problem you've identified is just as common with great mathematicians as well everybody else.