But by keeping those things together you're essentially cheating. You won't be able to remember these things in isolation as well. Also if it's easy for you to remember the rank of the matrix, but not the subspaces, you will be asked about the rank too often, since you connected it to subspaces.
In the example of the fundamental theorem of LA, the key point is the relationship between the subspaces (there are other ones just about the definition of a nullspace for example) - agree if every card was like this it would not be ideal.
I think my more important point was about problem sets, for instance:
Construct a matrix with the required property, or explain why you can't.
- left nullspace contains [1 3]^T
- rowspace contains [3 1]^T
breaks down into a few steps, and trying to break that down into bite sized flashcards such as, "what's the key idea in constructing a matrix with left nullspace [1 3]^T" and "given a matrix that is the product of L with row [1 3] can you choose a U so that combined they form a matrix with row space [3 1]?" seems like it could risk in resulting in me not being able to figure out the entire problem together. But maybe not?
Bringing it back to the post's proof example, what if you had completely nailed every step of the proof such that Anki doesn't ask you about it for a while, and then 6 months later, one of the harder steps comes up in isolation:
"What is the second adjustment we make in the proof of the ratio test, a < 1?"
what if you can't remember the first steps? Would having the rest of the context help? I guess it's a tradeoff.
Sure - for me, the cards slightly don't match the mental structure that is actually stored in my mind. In fact I remember the two steps "together", as this spatial move-and-squash. Perhaps I should rejig the cards.
