Great explanation! The argument seems to be more by contradiction though than induction. If you assume that m < n, then you replace all the blue vectors with red ones to form a spanning set in the way you described so crystal clearly. But then this gives a non-trivial linear combination of the m+1st red vector in terms of the first m red ones, contradicting linear independence. So m is at least n.
If you replace a vector other than w1 or w3 with v, the new collection might or might not span, depending on the particular vectors involved. But if you replace w1 or w3 with v, you are _guaranteed_ that the new collection spans, so that's what we need in the proof.
That's a separate little result to prove first---the key is that the coefficient of the vector you're replacing has to be nonzero, so that your new collection can still build v (just subtract off the other vectors and scale). This is a straightforward proof via the relevant definitions, along the lines of my "Span proof example" walkthrough on this same channel.
This is a very clear video - not even my Oxford Maths professor could explain it this clearly!
Thanks!
GOAT explanation
Thanks!
This video is incredible, thank you so much for such a neat proof!
this is the best video on exchange lemma. Thank you Frank !!
Perfectly explained, thank you so much! I was struggling with the proof of this my lecturer provided but this one made total sense almost instantly!
Thanks! Linear Algebra is a great class, and this is one of the highlights of the material!
A big Thanks from Germany :) super nice explained!
Danke schön!
Great explanation! The argument seems to be more by contradiction though than induction. If you assume that m < n, then you replace all the blue vectors with red ones to form a spanning set in the way you described so crystal clearly. But then this gives a non-trivial linear combination of the m+1st red vector in terms of the first m red ones, contradicting linear independence. So m is at least n.
Very clear explanation, good job!
great video, helped clear up some issues in my understanding; thanks
what happens if you try to replace any other vector but w1 an w3 with v?
If you replace a vector other than w1 or w3 with v, the new collection might or might not span, depending on the particular vectors involved. But if you replace w1 or w3 with v, you are _guaranteed_ that the new collection spans, so that's what we need in the proof.
but why does it still span when replacing with v?
That's a separate little result to prove first---the key is that the coefficient of the vector you're replacing has to be nonzero, so that your new collection can still build v (just subtract off the other vectors and scale). This is a straightforward proof via the relevant definitions, along the lines of my "Span proof example" walkthrough on this same channel.
@@frankswenton3177 thank you
a good video!! very clear!
Thanks, glad to hear it!
thank you so much sir
Glad you found it informative!
nice