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- เผยแพร่เมื่อ 29 ส.ค. 2024
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I love this series so much! Professor Benedict Gross once said "you can never study too much linear algebra" and I believe he is correct on that!
40:47 Based
Warm up problems are fun. For the first one I got that B was a basis for F_3 and not a basis for F_5. Also 4 = 1 and 5=-1 in F_3, so (4 5) is just a basis vector, so I think gamma_B (4 5) = (0 1) in F_3.
Proposition around @20:00 seems like just a restatement of the definition of linearly dependence: There's a vector of coefficients we call "alpha"... Scale it by the negative multiplicative inverse of a non-zero component then add that component's vector to both sides.
its worth noting that, with only two vectors you can tell if they are independent or not by checking if one is a multiple of the other. in fact, linear dependence is kind of just a generalization of "something is a multiple of another thing" but for when you have more than two things. thats probably why you didnt give any examples like that, because they arent as interesting, but for a topic so abstract, i think an example where you can just eyeball the answer intuitively is still instructive.
7:20 you say there are infinitely many solutions but I believe that only applies to infinite fields and since this field is finite there's is a finite amount. In fact since the space of solutions is one dimensional there are exactly 5 solutions.
He did this with a finite field before in a previous video, I think he is thinking of the solutions as "which real numbers have the correct remainder modulo p".
10:30 the best way I saw was to take the middle equation and subtract the other two from it. That neatly gets just -y=0 and it is clear from there.
The calculation at 16:14 is not necessary. Since the equation has to be true for all x, the coefficients of all powers of x have to be zero separately. And in this case the coefficients of the first and second power of x immediately deliver the result that the alphas have to be zero.
Someone in the comments of a previous video in the series explained that the vector space of polynomials should not be thought of as a vector space of functions, as they are not always equivalent objects. So I think your approach might strictly be the "correct" one.
23:24 For this argument to work I think one technically needs to prove that -1 != 0 for any field k. Which is pretty easy, but I'm not sure it was explicitly called out in this course yet. (Though I think we might have seen 1 != 0, which gets us there pretty easily.)
If we want to avoid proving that, we could use the fact that one of beta_i is nonzero, otherwhise v_i would be the zerovector, and it is easy to show that a set with the zero vector in it can't be linearly independent. But I agree, using the fact that -1=/=0 is easier.
1 != 0 is part of the definition of a field, 0 can't be the additive inverse of 1 since it's the additive identity, I don't think this really need to specifically be proven
@@TheEternalVortex42 There is one ring where 0 = 1, the set {0} with addition 0 + 0 = 0 and multiplication 0 * 0 = 0 . Could this not be considered a field as for multiplication we have closure, associativity, identity and inverses?
For a slightly simpler presentation of the first proposition's proof, could you say that since a reordered linearly dependent list is still linearly dependent, WLOG suppose that $ \alpha_n
eq 0 $, and then state that $ v_n = -1/\alpha_n ( \alpha_1 v_1 + \ldots + \alpha_{n-1} v_{n-1} ) $? (I'm just trying to check my own understanding here.)
I might be wrong, but this could work: th-cam.com/video/PvUrbpsXZLU/w-d-xo.html
P.S. A piece of advice: make video 1.5 times faster, I speak very slowly)