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The only way to test if you've 'understood' something is to apply it to real-world problems.

If it works - then you understood it. If it doesn't - then you didn't.

"Understanding" without way of external verification seems no different to dopamine-chasing.

Measuring on output and all that...

The only way to test is if you talk with an expert and he says you have understood. There are many things in linear algebra that you can use in practice even when you didn't really understand them. This is the reason why self-studying certain topics is very hard, you still need (good) teachers to give you constant feedback.

That seems like a verbalistic notion of 'understanding'.

How do I falsify the expert's claims about my understanding?

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