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Math Help - Strategy to Solve Proof Questions

  1. #1
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    Strategy to Solve Proof Questions

    Hi, I'm not sure how much help you guys can give, since this isn't really a textbook question...More of a "how should I study this".

    I find I'm able to do calculations and the like with ease in my linear algebra course, but it's the proofs that really get me. Every time I see a question that says "Show that..." or "Prove that..." I'm not able to figure out a method to complete the question. If I see the answer, I can understand just fine, but I don't understand how any person can think of that method to solve it.

    For example, one of the midterm questions I received:
    "If A and B are n by n matrices, and given that AB is invertible, prove that B must be invertible."
    I draw a blank when I try to solve this question on my own.

    When the solutions came up, I understood perfectly. To prove B is invertible, Bx = 0 must only have the trivial solution of x = 0. And then you start off with (AB)x = 0 and can easily go from there.

    So, are there some tips or strategies to get better at doing these questions?
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  2. #2
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    Quote Originally Posted by BlackBlaze View Post
    Hi, I'm not sure how much help you guys can give, since this isn't really a textbook question...More of a "how should I study this".

    I find I'm able to do calculations and the like with ease in my linear algebra course, but it's the proofs that really get me. Every time I see a question that says "Show that..." or "Prove that..." I'm not able to figure out a method to complete the question. If I see the answer, I can understand just fine, but I don't understand how any person can think of that method to solve it.

    For example, one of the midterm questions I received:
    "If A and B are n by n matrices, and given that AB is invertible, prove that B must be invertible."
    I draw a blank when I try to solve this question on my own.

    When the solutions came up, I understood perfectly. To prove B is invertible, Bx = 0 must only have the trivial solution of x = 0. And then you start off with (AB)x = 0 and can easily go from there.


    If you go onto take a "proof" course, this will become much clearer. Trying to learn how to prove things while taking linear algebra isn't easy
    So, are there some tips or strategies to get better at doing these questions?

    You just need to do a bunch of proofs most likely. Usually, for elementary proofs, you should write down everything that's given (everything you "know") and translate any definitions into their full meaning and then play around with things.

    There are some general tips for proofs, such as:

    If the proposition involves a negative, such as "there does not exist" or something along those lines, you should give contradiction a try

    If the proposition is of the form: if p then (q or r), you assume not q and try to prove r, (or assume not r and try to prove q)

    If the proposition states something like "there exists..." you're probably going to have to build/provide it to end the proof


    If you go on to take a "proofs" class this will become much clearer and easier. It is pretty challenging to try to learn the linear algebra and the proofs behind it at the same time because linear algebra courses dont normally focus of the method of proving things, like a proofs class would
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  3. #3
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    It's a real shame they jab you with a mandatory linear algebra course, and expect you to be able to learn and prove things at the same time. I don't think I'll go on to a proofs course unless I need to take more linear (although I might...)

    But I appreciate your advice. I suppose the only thing I can really do is study the theorems and think of all the properties that relate to what I'm being asked.
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by BlackBlaze View Post
    It's a real shame they jab you with a mandatory linear algebra course, and expect you to be able to learn and prove things at the same time. I don't think I'll go on to a proofs course unless I need to take more linear (although I might...)

    But I appreciate your advice. I suppose the only thing I can really do is study the theorems and think of all the properties that relate to what I'm being asked.
    It's not as hard as you think. You are asked to provide proofs because it is the only way to see if you have really understood what you learnt.

    Proving theorems is a skill which requires a good amount of practise. The best way to learn is not to learn the theorems by heart, but to pay very good attention to the proofs when you encounter them, and try to re-do them in your head later. Why is such and such true? What would happen if this part of the proof didn't hold? Why do we need this particular assumption?

    Proofs are what mathematics is about, and practise is the only way to get good at them.
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  5. #5
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    first, you have to understand the definition...
    and than try to prove the theorem that related with the topic a lot...

    remember, "practice makes perfect"
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  6. #6
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    Proofs are what upper level mathematics is all about, but for people who simply need to use the math as a tool in other fields, I don't think they should worry too much about the proofs. Of course it'd be great if we could all understand everything, but let the engineers and economists trust the mathematicians to make sure things work under certain conditions, then they can apply them
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