Instead of saying that an element of the tensor product /is/ a linear map, I'd say that V* tensor V is isomorphic to Hom(V,V). In other words, there's a way of interpreting a linear operator on V as an element of V* tensor V, and vice versa.
What do mathematicians have against examples. If a picture is worth a thousand words a math example is worth at least ten theorem's. No example always gets a dislike from me. Also my understanding is that contravariant things like vectors use the upper index and covariant things the lower index.
I agree there should be some examples. The coordinates of vectors are contravariant, and they indeed get upper indices, but the vectors in the basis then have lower indices to preserve the summation convention. So we might write the vector v = vⁱ eᵢ and the vⁱ then receive upper indices. This gets confusing since one is prone to conflate the vectors with their coordinates.
@7:25. "this gives a natural injection". I don't see it. What injection is the Prof talking about?? How? Still a mystery, 30 years later after first seeing the statement in Prof. Vraciu's tome... like Tantalus, forever denied understanding of that is right before me
well, I guess if you define a family of maps indexed by the vectors of V; where each map is from functionals to K by where the value of the map at functional alpha is the result of the evaluation operator using the indexing V-vector and alpha, that is the actual same set as the set of functionals of functionals -V**- (easy finite dimensional proof), voila, you baselessly indexed V** by V ("injected")
mathematicasn usualyy explain what they know they dont explain something to knew they only explain things that mathematicans already know but dont give explanation to introductory people which makes me think they speak alone
Hey! Thank you for this insightful video about dual vector spaces.
Really helpful video, I appreciate it, rather difficult to find videos on topics like this.
I'm glad you found it helpful. Let me know if there are other toppics you'd like to see...
super great material! i would be really happy having a lecture taught by Prof. Fowler on multilinear forms
Instead of saying that an element of the tensor product /is/ a linear map, I'd say that V* tensor V is isomorphic to Hom(V,V). In other words, there's a way of interpreting a linear operator on V as an element of V* tensor V, and vice versa.
One can see how studying this isomorphism grew into category theory. At least MacLean/Eilenberg 1948 used this example when introducing categories.
Heey Jim, very usefull video! But it would be very nice if you made a video with an concrete example, applying all this things. Thanks in advance
Very Very clear!!
What do mathematicians have against examples. If a picture is worth a thousand words a math example is worth at least ten theorem's. No example always gets a dislike from me.
Also my understanding is that contravariant things like vectors use the upper index and covariant things the lower index.
I agree there should be some examples.
The coordinates of vectors are contravariant, and they indeed get upper indices, but the vectors in the basis then have lower indices to preserve the summation convention. So we might write the vector v = vⁱ eᵢ and the vⁱ then receive upper indices. This gets confusing since one is prone to conflate the vectors with their coordinates.
would be nice to have a step by step explanation for why f* is the transpose of f.
It would be nice to have the transpose defined, but I guess he did it in earlier videos.
best video on dual space on UA-cam!
Glad you think so!
Great video
oh thank you! I'm glad you liked it.
Thank you so much, your video was so helpful :-)
How do you check that the dual basis you constructed is linearly independent, since this is needed to show that it spans?
thanks for this. only really needed the first half of the video though
really helpful video
thank you
You're welcome!
Good lecture. I would have started from the change of basis formula.
That's a good point.
Thnx hey, the video is so helpful
At 7:05, would the tensor product of V and V* be a linear map? You define this in your video for tensors as so.
@7:25. "this gives a natural injection".
I don't see it. What injection is the Prof talking about?? How?
Still a mystery, 30 years later after first seeing the statement in Prof. Vraciu's tome...
like Tantalus, forever denied understanding of that is right before me
"A map! of a map! of an "evaluation"?!? of a dual! applied to a MAP!!!"
"Sir, this is an Arby's"
well, I guess if you define a family of maps indexed by the vectors of V;
where each map is from functionals to K by where the value of the map at functional alpha is the result of the evaluation operator using the indexing V-vector and alpha,
that is the actual same set as the set of functionals of functionals -V**- (easy finite dimensional proof),
voila, you baselessly indexed V** by V ("injected")
What is a "natural" isomorphism?
Nicolas Diaz-Wahl I think it's one that does not depend on the basis you choose
great lecture but the background noise is disturbing
Yeah -- I made this video in grad school, and I definitely didn't have the best (or really any...) equipment.
Thanks man! This was useful!
mathematicasn usualyy explain what they know they dont explain something to knew they only explain things that mathematicans already know but dont give explanation to introductory people which makes me think they speak alone
More of variables with no real example
Sorry, but you make so much noise, the sound "mchch", when you speak. This is not good for a teacher!