demonstration/proof advice needed

inner08

Hello,

I'm presently taking a Linear Algebra I course (in French). I'm having difficulty learning how to do demonstrations using axioms/propositions.

For example, if I had to prove somcething like -(-B) = B or something like (B^-1)^-1=B, it just seems so obvious to me that its correct. I don't want the answer, I just want to learn how to basically prove simple problems like those so I can move on to harder ones.

I think the main thing I don't get is where do I start or what is the first thing I should do when I have demonstrations to do? (yes, I've looked at the list of axioms and such..)

Any insight would help!

Thanks,

matt grime

You have to start with the definitions. What is the definition of A^-1? it is the object that satisfies AX=XA=Id. So, what is B(B^-1)? It is Id, as is B^-1B, thus the inverse of B^-1 is B.

Take the definitions, and show that the thing in question satisfies the definition. This is why the most common piece of advice here is 'well, what is the definition of foo?' You'd be surprised (perhaps) at the number of people who don't know what it is that they are trying to prove and confuse that with the inability to prove it.

HallsofIvy

What makes you think it is "obvious"? Because every one knows that (-1)(-1)(B)= B? That's missing the point. "-B" does not mean (-1)B. "-B" means "the additive inverse of B". In a group, every member has an additive inverse so given B, -B exists. And, then, it must have an additive inverse. What is the additive inverse of -B?

Presumably, your definition of additive inverse is something like: For any X in the group, -X is the unique member of the group such that X+ (-X)= 0 (the group identity) and (-X)+ X= 0. So the additive inverse of -B (call it "A" for the moment) must satisfy (-B)+ A= 0 and A+ (-B)= 0.

On the other hand, if you are saying it is obvious because of the "obvious" similarity of those equations defining the additive inverse, you are completly correct. Just write that down as clearly as you can!

Edgardo

Hello inner08,

my advice would be to take a calculus or linear algebra book and follow some of those easy proofs. The important thing is to recognize which of the axioms is used in each step. You will then recognize tricks such as adding a zero.

As mentioned above, you will learn about tricks such as adding a zero. If you don't know where to start it can help to work "backwards". First look which result you want to get, then work backwards.

Remember, in each step you have to be aware which axiom in the list you are using.

After you've followed some of the proofs, try to prove the statements yourself the next day.
Start with the easy ones.