The basis of a vector space part 2 -- Abstract Linear Algebra 11

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[Spacexioms]

I hope you can do this full time in the future. This is top tier quality education

[StratosFair]

The proof of theorem 19:00 is nice, but a bit tedious. Wasn't there a way to argue by contradiction and using what we have shown before about minimal spanning sets ?

[schweinmachtbree]

perhaps it should be pointed out that the theorem at 19:00 only applies to finite-dimensional vectorspaces, whereas almost all of the other theorems so far can be adapted to possibly-infinite-dimensional vectorspaces (using the definitions that the span of an infinite subset is the set of all *finite* linear combinations, or equivalently the union of the spans of all its finite subsets, and that an infinite subset is linearly independent if all of its finite subsets are linearly independent)

[synaestheziac]

That hat notation is so much nicer than writing in the i-1 and i+1 terms!

[BethKjos]

@4:46 first proof could be as simple as this: if w is other than zero, then B isn't maximal; contradiction and done.

[StanleyDevastating]

I don't understand the point of the theorem at @19:00 . We already proved that a set being maximally linearly independent was equivalent to a set being the minimal spanning set. So we already know that a set of linearly independent vectors cannot have more members than any set that spans the vector space, right?

[schweinmachtbree]

The proof of the final implication (3) => (2) in the first theorem is incorrect. the proof of the first implication (1) => (3) is also strange (maybe not incorrect, but definitely strange) - one doesn't need to bother with the alphas; one can just consider B U {v} from the start and follow ones nose (B U {v} is dependent so there is a relation lambda_1 v_1 + ... + lambda_n v_n + lambda v = 0. lambda must be non-zero so we can divide by it and rearrange to find that v in span(B) as was to be shown)

[l.p.]

Great lecture! For what it's worth it seems like YouTube maths loves dimensions and infinite cardinalities, could clickbait with that in the title.

[SurfinScientist]

So what is the basis of the vector space defined by x_1 + x_2 + x_3 = 1?

[jeffreycloete]

Hi Prof Penn..Thanks for your continued efforts to bring maths to a larger audience..could you please do a video on EXACT SEQUENCES..maybe ..SHORT EXACT SEQUENCES..in the same vein as your wonderful and very popular video on Tensors?

[artificialresearching]

P.S. A piece of advice: make video 1.5 times faster, I speak very slowly)