Overcoming Challenges with Self-Study: Struggling to Improve

[bonfire09]

One of things I noticed when I self study is when I go check my answers against the solutions, some of my answers seem to be way off. For example there was a question that went "show that for any nxn non singular matrix it is row equivalent." I happened to show a proof by induction but in the solutions it just drew a arbirtrary nxn matrix with 1's running down the diagonal( these entries are obviously the pivot positions). I'm not sure if this normal or not?Will I just get better by practicing and struggling? But I fear I won't be able to get any better than I am now. The way I approach each section of a book is I first read very carefully and reread a few times before I feel I have a good understanding. Then I go to the problems and try to do all of them on my own. About 3/4 of them I can do and end up with the correct answer. Of course I recheck my solution. Then I check my solutions against the solution manual but at times like I said earlier my answers are way off than what is written in the solutions. This happens mostly on problems that say "show this...". I try my best to go back and fix my answer but I just end up leaving it since I feel its not worth it since I already know the solution. I'm trying hard to make sure that I can get up to the point where I can answer all the problems correctly but it seems quite hard.

 

[AlephZero]

There are often several ways to prove the same math result. The fact that your proof is completely different from the book doesn't necessarily mean you are wrong.

If you want advice on whether your proof is correct, or help understanding a proof in the book, the math forums here are a good place to ask!

 

[bonfire09]

Of course I ask for help on these forums but I don't want to ask for help on all the problems I'm stuck on here. I just feel that if can't do all the problems in each section that I don't understand the section well then. And once I know the solution I feel its useless going back since I already know the answer. And if I don't check my answers then ill never know if I did it correct or not. I do make an honest effort on each problem and at times like I said my answers are way off and I get discouraged easily. Hopefully you understand where I'm coming from.

 

[Robert1986]

Of course I ask for help on these forums but I don't want to ask for help on all the problems I'm stuck on here. I just feel that if can't do all the problems in each section that I don't understand the section well then. And once I know the solution I feel its useless going back since I already know the answer. And if I don't check my answers then ill never know if I did it correct or not. I do make an honest effort on each problem and at times like I said my answers are way off and I get discouraged easily. Hopefully you understand where I'm coming from.

I think you need to keep in mind that you are just an undergraduate student taking (I'm guessing) linear algebra. That is, you are still pretty young math-wise. The solutions are probably written by a graduate student or the author of the textbook. It is only natural that the solutions in the manual will be much more elegant (though that might not always be the case) than yours, and like AlephZero said, there are many ways to prove something.

And the same thing goes for the computational problems.

 

[bonfire09]

Oh ok thanks. Now I'm encouraged to go on and struggle it out.

 

[3.141592]

And once I know the solution I feel its useless going back since I already know the answer.

When I was learning logic, I was always given the premises and the conclusion. The problem was to show what steps would lead from the premises to the conclusion, having at the end discarded any assumptions brought in along the way for help. That always made me laugh a bit when I'd get stuck (a lot) - I have all the data and am even given the solution; all I have to do is build the bridge and yet here I am stuck!

So if you turn to the solution and yours is wrong, I encourage you to retry your attempt. Often I find that seeing the solution makes me go, 'Oh ok!' and recognise where I went wrong. But if not, think of it this way: you now have one extra piece of data for making your solution. You still need to build the bridge from the data in the question to the solution provided. Plus, you can now work backwards from the solution to the question and maybe meet somewhere in the middle!

Your confidence will grow when you go back and spot where you went wrong, I think. It is very disheartening to see you have got it wrong but not see why.

 

[homeomorphic]

And if I don't check my answers then ill never know if I did it correct or not.

You have to try to move beyond that. You can never be 100% sure you are right about anything, but you should get to a point where you can determine for yourself whether your answer is correct, ignoring the occasional oversights, which can usually be eliminated by double-checking repeatedly. In theory, you can reduce everything down to the basic logical deduction, modus ponens, p and p implies q, therefore q, which isn't really that complicated of a thing to check. It's always the same form, but you have all kinds of specific p's and q's. In practice, we don't do things that formally, but still...

 

[bonfire09]

Thanks guys and I agree with 3.141592's answer plus I'm trying to self study these things on my own in absence of a professor or friends. I guess that I should change the way I think. I shouldn't take it hard upon myself if I get some my answers wrong. Eventually i'll get better even though it will take me some time on my own. Most of us who are self studying I guess go through a similar process like I'm going through and eventually we all get better unless of course you are a genius.

 

[Jorriss]

In practice, we don't do things that formally, but still...

When I get stuck on a problem, I do do that. I make sure every single step is justifiable entirely and then I know my final result is correct. It may be utterly unworkable and a useless result, but correct.

 

[3.141592]

Thanks guys and I agree with 3.141592's answer plus I'm trying to self study these things on my own in absence of a professor or friends. I guess that I should change the way I think. I shouldn't take it hard upon myself if I get some my answers wrong. Eventually i'll get better even though it will take me some time on my own. Most of us who are self studying I guess go through a similar process like I'm going through and eventually we all get better unless of course you are a genius.

I'm studying by distance learning so only have access to a tutor to mark my assignments and by email. It's pretty annoying. So I share your pain. Plus, learning on your own usually makes you feel either you're a genius or a moron. In both cases you'll be wrong.

When you get a solution wrong, try it with another easier method. When you get it right, try it with another harder method.

A simple example: when converting units in a calculation let's say involving division, it is easy to cancel just by drawing a line through whatever like terms are written above and below the division line. It's a tiny bit harder to move units above or below the division line, and adjust the + or - sign of the exponents accordingly, and then add or subtract powers. Mostly because you have to do a tiny bit more thinking and there is more chance for arithmetical errors.

But if you get it wrong, try it the easier way and see if you get it right. That might open up the door to where you went wrong the first time. And if you get it right, trying it the harder way might reveal some mechanism that was hidden with the easier method, or just generally help you get a different POV on the problem.