>> 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.
>No disagreement here. Perhaps Thomas only meant to warn against >blind memorization without understanding. This way a student would >end up with the ability to repeat the memorized definition but >nothing else. This danger is even greater, when those definitions >are memorized as isolated entities, without checking them against >examplex, counterexamples and proofs.
I agree. After you have those definitions memorized, you have to get understanding of those definitions through theorems and examples, which show unforseen consequences of those definitions.