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:
The calculation type question:
The weaker calculation type question:
The weaker prove type question:
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
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:
Quote:
Prove the triangle inequality using Cauchy-Schwarz inequality and then generalize the theorem to complex Euclidean space |
The calculation type question:
Quote:
Diagonalize the matrix ##A## and find its eigenvector. |
The weaker calculation type question:
Quote:
Verify that equations (1.2) and (1.3) get (1.4) |
The weaker prove type question:
Quote:
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
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