MHB Solve Linear Algebra Exercises: Tips & Techniques

[caffeinemachine]

Here's something that's been bugging me for quite some time.

I have been reading Linear Algebra Done Right by Axler. By the time I reached to the exercises at the end of chapter 5, I started realizing that I was not able to solve the exercises "from scratch". The exercises seemed trivial if I used some key results from the chapter but when I tried to think naturally and "originally" I either was not able to get anywhere or I discovered some fantastic new (usually long) solution at the cost of spending a lot of time on the question; where the former occurred more frequently. (And this also goes for Abstract Algebra and other mathematical disciplines).

Now here's my question. What according to you is the right way to do it? Mugging a few results makes things a lot easier that just tackling the question with having zero prior knowledge.

How do you guys do it? How much do you think your previous problem solving experience matters to you? How much do you depend on theorems or results you have previously read?

 

[Fantini]

In general, it's expected for you to use those key results when doing the exercises. They may be trivial, but you get at least some practice in applying them in different contexts, whereas doing things "from scratch" is good when you already have a decent grasp of the tools at your disposal. It takes time and great efforts, but you also acquire great experience in thinking differently. I just believe it's best used once you've mastered the usual knowledge, else you're missing out on a lot.

Remember that "originality" comes in many forms: solving problems without pre-existing tools, using pre-existing tools for a completely different subject, using pre-existing tools in a very innovative way, creating tools tailored to your problem and finding out it applies to several others. Don't be too hard on yourself: it's fun to fight "left-handed" sometimes, but don't let it become a severe restriction to your learning.

As for me, I have no qualms to using results I know, wherever they come from. My previous problem solving experiences taught me that I often tackle "ill-equipped" or underusing techniques, so I decided to keep as much as I can at hand.

Bear in mind this is all my honest opinion and it might not really match what you had in mind.:D

 

[caffeinemachine]

In general, it's expected for you to use those key results when doing the exercises. They may be trivial, but you get at least some practice in applying them in different contexts, whereas doing things "from scratch" is good when you already have a decent grasp of the tools at your disposal. It takes time and great efforts, but you also acquire great experience in thinking differently. I just believe it's best used once you've mastered the usual knowledge, else you're missing out on a lot.

Remember that "originality" comes in many forms: solving problems without pre-existing tools, using pre-existing tools for a completely different subject, using pre-existing tools in a very innovative way, creating tools tailored to your problem and finding out it applies to several others. Don't be too hard on yourself: it's fun to fight "left-handed" sometimes, but don't let it become a severe restriction to your learning.

As for me, I have no qualms to using results I know, wherever they come from. My previous problem solving experiences taught me that I often tackle "ill-equipped" or underusing techniques, so I decided to keep as much as I can at hand.

Bear in mind this is all my honest opinion and it might not really match what you had in mind.:D

Seems like you've been through the painful transition phase I am currently going through. Thanks man, your post really helped.

Other math wizards, your opinions are welcomed too.

 

[Swlabr1]

Here's something that's been bugging me for quite some time.

I have been reading Linear Algebra Done Right by Axler. By the time I reached to the exercises at the end of chapter 5, I started realizing that I was not able to solve the exercises "from scratch". The exercises seemed trivial if I used some key results from the chapter but when I tried to think naturally and "originally" I either was not able to get anywhere or I discovered some fantastic new (usually long) solution at the cost of spending a lot of time on the question; where the former occurred more frequently. (And this also goes for Abstract Algebra and other mathematical disciplines).

Now here's my question. What according to you is the right way to do it? Mugging a few results makes things a lot easier that just tackling the question with having zero prior knowledge.

How do you guys do it? How much do you think your previous problem solving experience matters to you? How much do you depend on theorems or results you have previously read?

The point of Linear Algebra Done Right is to get you to think about Linear Algebra in a very different way from "usual" Linear Algebra texts. So, don't just do the questions - you want to do them they way they the author would do them! (That is, if you ever use a determinant in an argument, then there is probably a different way...)

 

[caffeinemachine]

The point of Linear Algebra Done Right is to get you to think about Linear Algebra in a very different way from "usual" Linear Algebra texts. So, don't just do the questions - you want to do them they way they the author would do them! (That is, if you ever use a determinant in an argument, then there is probably a different way...)

Yeah that's right. I was not talking especially about Axler's book. Mentioned it just as an example.