It’s particularly annoying that nobody gave me an answer, because the answer turns out to be simple
The answer you give might be the simple answer to the question you asked, but I have trouble figuring out what you were suggesting in the first place (even given the answer!). Figuring out the question, the confusion, so that it’s possible to supply the simple answer is really hard.
Similarly, I’m skeptical of your claim that your calculus text failed an easy chance to communicate insight. I’ve had a lot of bad experiences with textbooks, where I eventually figure it out, perhaps from another source, come back and can’t see what was wrong with the book. If there is something worthy of indignation for its not being shared, why don’t you share it?
Let me try, and likely fail, to communicate mathematical insight: matrices are evil. Moving to matrices involves choosing a basis. Usually, as in Scott’s example, you just want a direct sum decomposition; it’s more natural, and it doesn’t clutter the problem with unnecessary entries or indices.
It’s particularly annoying that nobody gave me an answer, because the answer turns out to be simple
The answer you give might be the simple answer to the question you asked, but I have trouble figuring out what you were suggesting in the first place (even given the answer!). Figuring out the question, the confusion, so that it’s possible to supply the simple answer is really hard.
Similarly, I’m skeptical of your claim that your calculus text failed an easy chance to communicate insight. I’ve had a lot of bad experiences with textbooks, where I eventually figure it out, perhaps from another source, come back and can’t see what was wrong with the book. If there is something worthy of indignation for its not being shared, why don’t you share it?
Let me try, and likely fail, to communicate mathematical insight: matrices are evil. Moving to matrices involves choosing a basis. Usually, as in Scott’s example, you just want a direct sum decomposition; it’s more natural, and it doesn’t clutter the problem with unnecessary entries or indices.