How do I improve my proof writing style?

[TachyonLord]

So I've taken this Linear Algebra class as an elective. So there's stuff that is so obvious and logically/analytically easy to prove but I honestly don't understand how to prove them using the standard way. So what should I do about this ?
And I really like linear algebra so I don't want to mess this up.
Thanks.

 

[jedishrfu]

Talk to your prof about it.

In high school geometry we would write two column proofs.

The first column was the assertion like AB and BC line segments are perpendicular and then column two would say why or what axiom or theorem we invoked.

Theres the book of proofs inspired by Erdos but it covers a range of proofs and they ate the best if the best.

Get a good book on set theory and read and do the proofs.

 

[TachyonLord]

Any other suggestions ?

 

[jedishrfu]

You could also work with your classmates challenging one another to find a flaw in your proof.

Mathispower4u.com has some videos on Linear Algebra:

http://www.mathispower4u.com/linear-alg.php

Lastly, Schaums Outlines has a linear algebra volume that you could look at.

 

[PeroK]

Any other suggestions ?

It's fairly common to meet formal proofs for the first time in linear algebra and to find it a new and difficult way to think. It's also fairly common to find that things that seem very obvious are awkward to prove rigorously.

If that's the case, then you need to recognise that this is a new way of thinking, don't be too worried that it's difficult to prove easy stuff and try to focus on improving and understanding logical thinking.

It would be better if you posted an example, so we can see better where you are finding difficulties.

 

[jedishrfu]

Heres a brief set of LA proofs with solutions. You might others online.

http://www.math.ucdenver.edu/~esulliva/LinearAlgebra/Proofs.pdf

Try proving it yourself and then how it was done.

Pay attention to steps you think are trivial as there may some subtlety that you didnt consider.

 

[WWGD]

Any other suggestions ?

Try explaining them to people, specially those that have slightly less training than you do, but know enough to understand. It will force you to clarify your ideas, which will reflect in your writing.

 

[WWGD]

Heres a brief set of LA proofs with solutions. .

I love LA!

 

[mathwonk]

always begin with the definition of the relevant term. e.g. to prove that a certain collecion of vectors is linearly independent, recall first the definition of liner independence.

e.g. to prove that the sequence (1,0,0), (1,1,0), 1,1,1) is linearly independent, (over the rationals say), recall that a sequence v1,v2,v3 is linearly independent over Q if and only if the only rational scalars a1,a2,a3 such that a1v1 + a2v2 + a3v3 = 0 (the zero vector), are the scalars a1 = a2 = a3 = 0.

Then phrase your problem in these terms: i.e. hence to prove that our sequence (1,0,0), (1,1,0), 1,1,1) is linearly independent over Q we must show that the only rational numbers a1,a2,a3 such that a1(1,0,0) + a2(1,1,0) + a3(1,1,1) = (0,0,0), are the rationals a1 = a2 = a3 = 0.

Now go to work calculating with your example: i.e. we are assuming that a1(1,0,0) + a2(1,1,0) + a3(1,1,1) = (0,0,0), and we want to deduce that a1 = a2 = a3 = 0.
If we calculate we get starting from our assumption, that (0,0,0) = a1(1,0,0) + a2(1,1,0) + a3(1,1,1) = (now calculate) (a1,0,0) + (a2,a2,0) + (a3,a3,a3)
= (a1+a2+a3, a2+a3, a3). Now look at your calculation and try to see how to deduce your result:

I.e. since we have (0,0,0) = (a1+a2+a3, a2+a3, a3), looking at the last entry we see that a3 = 0. Substituting that value gives (0,0,0) = (a1+a2, a2, 0). Now looking at the second entries gives a2 = 0. Substituting again gives (0,0,0) = (a1,0,0), hence a1 = 0 also. Hence we have a1 = a2 = a3 = 0.I have written this proof for the absolute beginner, who needs every step spelled out from definition to conclusion. If you are writing for more experienced readers, you tend to skip more and mopre of these steps, but this is the way to begin.

But to re-iterate: a proof is a logical argument leading from a definition to a specific conclusion, hence it is always prudent to begin by stating, then specializing, the relevant definition involved. I.e. you cannot prove the truth of a statement involving technical words unless you know exactly what those words mean.

 

[FactChecker]

Make sure that every "obvious" statement is the direct conclusion of an already proven theorem, lemma, or axiom. Then use that theorem/lemma/axiom in the proof, after showing that every requirement is met. This takes some practice and the first course of linear algebra is a good place to start. Only after the proper habits are well established (to everyone's satisfaction) can some steps be skipped in the writing (not in the thinking).

 

[TachyonLord]

Thank you so much everyone ! :) I'll see to all of them !