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- เผยแพร่เมื่อ 13 ก.ย. 2024
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The first proposition forgets the condition on the right side of the that W needs to be non-empty. This condition is needed in the proof of the
21:26 I think it'd be nice to show that if S is a subset of V then span(S) is the SMALLEST subspace of V containing S
3:33
technically you also have to make sure that the subvector space (V',+) 's operation agree with your vector space (V,*).
that is they have the same field equipped K and u+v = w iff u*v=w given that u and v are in V' (and so w as well).
I agree but I think the operations are clear from the context of the statement.
27:17
lol
Related to the last warm-up problem, I got curious and found out that the number of possible vector subspaces over a finite field is related to stuff involving the q-Pochhammer symbol, which you covered in the number theory playlist
isnt the last warm up example impossible for F3? if your only elements in the field are {0,1,2}, how can you possibly add to 9? i guess the answer is "no, its not a subspace", but then why am i asked to describe it geometrically?
9 = 0 in F3
At 16:20-ish it might be worth emphasizing that the limit of polynomials or degree ≤ N is the set of all polynomials and not the set of power series. It’s a misunderstanding of the concept of limits that I’ve seen students fall into more than once.
What is the difference exactly? Is it that with polynomials you have infinitely many powers of x to choose from (the basis 1, x, x^2 etc. has infinite elements) but any concrete polynomial only consists of a finite linear combinations
and with power series you have in a sense an infinite polynomial or rigorously speaking a power series is the limit of sequence of polynomials.
Am I right with this interpretation?
Your way of explaining things makes it so easy to follow your videos, thank you for that. However I realized that even though I mostly understand the ideas and concepts it is hard for me to remember and connect them together, is there some way of thinking or style of lerning I could employ to better memorize them?
I think of these structures like a pyramid, less rigorous structures are smaller building blocks and their features combine until you reach vector space or algebra. Try writing down a flow chart with these structures and the properties that you add/subtract to build them. Similarly for transformations write down the key elements of a mapping and their features to make a function, objection, linear transform, etc
Another way to learn this stuff is to do lots of problems. At least this helped me a lot. I know it is hard to motivate yourself to do some exercises, so some sort of course you can follow and do weekly assignments could help. The vocabulary and concepts you will eventually know by heart just because you used them so often
@16:30 I was trying to think why W was a subset if W is all functions that have f(3) = 0, of course it must be only the set of all differentiable functions that have f(3) = 0. lol
Related to the final proof: for all subspaces W of V, could we find a set of vectors that span W?
Yes, trivially, just take the whole set W then the span of the whole set is W.
@@jimallysonnevado3973 and the smallest set of vector for the span?
@@joel.9543 The smallest set of vector to span a space is called a basis of that space. There is a theorem that says evey vector space (a subspace of a vector space is trivially a vector space) has a basis, but it requires Zorn's Lemma (a variant of axiom of choice). However, I think for finite dimensional vector space, axiom of choice is not needed but I'm not very sure.
Generally speaking for a subspace W of dimensions N you can choose N linearly independant vectors in W, and that will span W. If you lose any dimensions in the span you didn't choose linearly Iinependant vectors, and if you spanned into too many dimensions you broke the subspace closure with one of the vectors you chose (so it wasn't from W or W wasn't a subspace). But there are usually going to be a lot more ways to chose span than there are subspaces, bear In mind of all the possible linear combinations of vectors from W with coefficients from V a lot of them will be linearly dependant, and generate the same subspace. For example, in R3 any plane through the origin is a subspace, and any 2 non-parralel vectors in that plane will span the plane, as you could write any vector in the plane as a linear combination of them. So you could choose 2 orthanagnal vectors with a unit length, or you could choose 2 vectors that have an arbitrarily small interior angle and differing magnitudes, and their span will still generate the same slice in R3
The short answer, I suppose, is: yes we can find these, they're called a basis, and they're rather important, so I'm certain they will be featured in an upcoming video in the series...
Why are you denoting the scalar field as "k" instead of the traditional "F"?
_k_ is used in quite a few textbooks - especially older ones. In any case, it was well-defined to be a field, so I don't think there's a risk of confusion. In most circumstances, Algebra proofs don't bother to mention the field over which the vector spaces hold; it tends to be implicit for scalar fields _k^1_.
It is mentioned in I think video 2 or 3
Professor Penn, have you considered fadeouts after "That's a good place to stop"? Might make the end of videos not end with you blurred and putting the chalk back on the blackboard ledge, etc. Love this course BTW.
but W’ is really just the trivial subspace, since e^x is the only (class of) functions that is its own derivative, and it’s never 0 at 3…..
Why you think that W' must have f(3)=0?