>>a significant role. They often stumble because (among other problems) >>they don't realize they need to _memorize_ definitions _precisely_.
>Memorize? I can't remember ever memorizing anything. Better just to >practice until you understand. Discuss, ask questions, apply. That way >you memorize, of course, but that's just a side-effect.
No, for proof classes, which linear algebra is in many places, it is crucial to *memorize* the definitions. This holds for all the proof classes at higher levels also. There is simply no way of giving rigorous proofs if you don't know the actual definitions. All too often, students have some very vague ideas of what is going on and then can't even get started on a proof because they don't know the *exact* definiton used in the course.
>>So you can do a little self-assessment here to figure out whether >>what you're missing is bits of topics or the core idea: can you, >>right this minute, define what a vector space is?
>Hopefully not just as a one-to-one rendition of phrases from a >textbook...
The student should be able to give a rendition that is at least equivalent to the one given in the book and that uses precise language. If you can't say that a basis is an independent spanning set then you don't know what a basis is. If you can't give the quantifiers for the definition of independence, you won't be able to do a proof using independence.