Criteria for instructive problems in self-study

[Seydlitz]

Hello guys, I hope you all have happy holiday!

This question crops up to my mind often when I read through threads in PF. One of the most common points for books recommendation is because of good and instructive problems in them. For specific example, WannabeNewton in this thread is very concerned on having "good" problems in the book. My question is this, as suggested by the title, what are the criteria for instructive problems? How do you know which problems are worth doing and which should be skipped or reserved for second reading? Especially when you are considering this in self-study context where there is no instructor available or problem sheet assigned.

To add further background to my question, I'm currently working through Mathematical Methods book by Mary L. Boas, and I'm currently in eignvalues and eigenvectors sub-section of linear algebra chapter. If you have the book, you can see already that there are about 60 questions overall there. I usually do most of the exercise questions because they really help me to understand the topics but in this case I want to speed up my progress a bit because I would like to touch the chapter on Differential Equations before the university began.

Here are some problem examples that illustrate the type of the problems found in the book and also other books that I've encountered. (The problems are not written verbatim in this post.)

The prove type question:

Prove the triangle inequality using Cauchy-Schwarz inequality and then generalize the theorem to complex Euclidean space

The calculation type question:

Diagonalize the matrix $A$ and find its eigenvector.

The weaker calculation type question:

Verify that equations (1.2) and (1.3) get (1.4)

The weaker prove type question:

Show that an orthogonal transformation preserves the length of vectors.

Certainly these type of problems are not only found in one particular book but also in other mathematical books (even in Calculus by Spivak). Which type of questions then should one focus more according to you?

Thank You

 

[AlephZero]

I'm not sure that your attempt at a classification system is very useful here.

IMO the "best" questions are the ones which give you more insight into what the concepts mean, and "what is going on". Of your examples, the "best" question might be the last one, the insight being the orthogonal transformations behave like geometrical rotations in that they preserve lengths.

The second one could be a rather pointless plug-and-chug exercise (and in real life you would use a computer to do it, not hand calculation) - but if matrix A had some special properties like repeated eigenvalues, or an incomplete set of eigenvectors, you might get some valuable insight from working through the details by hand.

 

[R136a1]

In my opinion, the proof-type questions are the most valuable type of question. They really test your understanding of the concepts.

The computation type questions are of course still important. You should know how to find eigenvalues and diagonalizations. But once you solved a couple of them and the process is clear, then making more of them seems less useful. While proof-type questions are always valuable to me.

If I read a math book, I always do some computational stuff first to get my hands dirty. But when I feel I know what's going on, then I start doing the proofy stuff and I spend way much time on that.

 

[Seydlitz]

I'm not sure that your attempt at a classification system is very useful here.

IMO the "best" questions are the ones which give you more insight into what the concepts mean, and "what is going on". Of your examples, the "best" question might be the last one, the insight being the orthogonal transformations behave like geometrical rotations in that they preserve lengths.

The second one could be a rather pointless plug-and-chug exercise (and in real life you would use a computer to do it, not hand calculation) - but if matrix A had some special properties like repeated eigenvalues, or an incomplete set of eigenvectors, you might get some valuable insight from working through the details by hand.

Ah okay I get what you say, that the most interesting problems are the one that show you some important concept of the chapters. I'll definitely keep this in mind.

The most noticeable fact that I see from the matrix in the exercises is they are always symmetric or Hermitian.

In my opinion, the proof-type questions are the most valuable type of question. They really test your understanding of the concepts.

The computation type questions are of course still important. You should know how to find eigenvalues and diagonalizations. But once you solved a couple of them and the process is clear, then making more of them seems less useful. While proof-type questions are always valuable to me.

If I read a math book, I always do some computational stuff first to get my hands dirty. But when I feel I know what's going on, then I start doing the proofy stuff and I spend way much time on that.

Because I'm learning how to do proof myself, I see these type of questions as rather interesting and challenging. I'll do this also in the process.

 

[thelema418]

Self study depends on intrinsic motivation. What do you wish to accomplish?

Personally, I prefer to read articles or academic texts. When I find terminology or reasoning that I don't understand or that I feel unfamiliar with, I note it and either try to research it or solve it independently. With respect to linear algebra, I found it more helpful to examine the concepts as they came up in differential equations rather than anything I've done as problems in a linear algebra textbook.

For proofs you may want someone who can be critical to review your work. It may be easier to justify something to yourself than a critical friend.