Tips for proof based linear algebra?

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To prepare for a final linear algebra exam focused on proofs, extensive practice with challenging problems is crucial. Mastery of definitions is essential, as precise language is necessary for constructing valid proofs. Reviewing existing proofs helps in understanding how definitions are applied in context. Engaging with a variety of proof techniques will enhance problem-solving skills. Consistent practice is the key to success in mastering proof-based linear algebra.
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What would be a good way to prepare for a final linear algebra exam (vectors, planes, matrices) where every question is asking for a proof?
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Practice. Do a lot of problems, especially hard ones. There is no other way.
 
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Absolutely essential- learn the definitions! And, by that, I don't mean "get a general idea of what is meant". In mathematics, definitions are "working definitions". You use the precise words of the definition in proofs. So read a number of proofs, observe how they use the words in definitions and previous proofs in the proofs themselves. Then, as micromass said, "practice, practice, practice"!
 
Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
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