Replacement Theorem Proof

Proof of the replacement theorem, one of the most important theorems in linear algebra. It intuitively says that any linearly independent set can be extended to be a spanning set. It is used, for example, to show that the notion of dimension is well-defined. The proof is absolutely beautiful and requires use of the intruder theorem. Enjoy!
Replacement Theorem Video: • Replacement Theorem
Intruder Theorem Video: • Intruder Theorem
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[@anzuluv]

Thank you so much for making this video online linear algebra is rough.

[@MrCigarro50]

Como siempre...fantástico. Gracias Dr. Peyam.

[@mikewagner2299]

Correct me if I missed it, but doesn't this proof need one more rather trivial case? I believe this is only valid for M

[@drpeyam]

It’s basically how the set is defined; by definition of m = n, the set is the empty set

[@mikewagner2299]

@@drpeyam I thought about it again and there is technically enough math to extend it to M=N but there wasn't an explicit statement. Your inductive proof was valid from M€[0,N-1] however because of that, M+1 or M=N is also valid

[@lrequiemalbertozamora7570]

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[@mukeshchand5301]

Love from India

[@sumitshingare7484]

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[@harrytaylor4360]

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[@kennybanteinamkharmujai1548]

Thank u so much sir. 💥💥💥💥💥

[@DP-sq7lw]

Very subtle, but great explanation. Thank you

[@drpeyam]

Thank you!!!

[@marcialvalero622]

You are so cool!!!

[@drpeyam]

Thanks so much :)))

[@jonshonjohn4756]

First!