Those Functions f are just transposed Vectors: Linear Transformations "are" Matrices, so LT from n Dimensions to 1 Dimensions are just 1×n-Matrices, that is, row vectors. So the elements of V* are Scalar multiplications with a vector in V. So (V*) * are Linear Transformations, that is, Matrices, taking those row Vectors and turning them into a Scalar. Those must therefore be Column vectors once again, the elements of V. Therefore V** is V
Yes! I was waiting for this one! I remember in an interview with you and bprp and maybe Chester I can’t remember you said in a video you would do V*****!
so i believe, that the gist of the video is that f(x) ≡ fx ≡ (f)x, where x ∈ V and f ∈ V*, can be interpreted twofold: either as a functional f(⋅) ∈ V* := L(V, F), i.e. a linear function f(⋅): V --> F, which evaluates f on every vector in V, or as an evaluation map (⋅)x ∈ V := L(V*, F), i.e. a linear function (⋅)x: V* --> F, which evaluates every functional in V* at x. correct? ;)
V(ascot(balmoral(baseball(beanie(bearskin(beret(bicorne(bowler(chullo(cloche(cricket(sombrero(....(x))...) is a lot of dual hats (partially listed from wikipedia)
@@drpeyam hahaha no need to thank me. You are just naturally a likeable person. I like to say 'earth is lucky to have you'. It is especially true about you!
The day you'll stop smiling we'll stop watching your videos! Your enthusiasm is the key! Thanks again!!!!
* * * * *
Five stars for Dr.Peyam!
Your dual spaces playlist is fantastic.
Still hope you'll extend it to infinite dimensional vectorspaces one day.
Press a gigantic F to pay respect to this amazing teacher
Not going to lie, I did watch the whole video for the entertainment purpose...
no idea what the ***** it was....
Hahahahaha
Ok, Peyam! This is too much for me!
Those Functions f are just transposed Vectors:
Linear Transformations "are" Matrices, so LT from n Dimensions to 1 Dimensions are just 1×n-Matrices, that is, row vectors. So the elements of V* are Scalar multiplications with a vector in V.
So (V*) * are Linear Transformations, that is, Matrices, taking those row Vectors and turning them into a Scalar. Those must therefore be Column vectors once again, the elements of V. Therefore V** is V
ayyy , cool interpretation
what an insightful comment!
Your so great! Thank you for your enthusiasm and work
Yes! I was waiting for this one! I remember in an interview with you and bprp and maybe Chester I can’t remember you said in a video you would do V*****!
and it turns out * is just a fancy version of ^-1 all along.
As a self-learner of linear algebra, dual space part was discouraging and your video helped me. insightful comments from experts are also helpful.
Awesome👍👍👍👍👍
I feel like I'm going to wind up learning things like automorphism groups.
OK. Thank you very much.
That escalated quickly
so i believe, that the gist of the video is that f(x) ≡ fx ≡ (f)x, where x ∈ V and f ∈ V*, can be interpreted twofold:
either as a functional f(⋅) ∈ V* := L(V, F), i.e. a linear function f(⋅): V --> F, which evaluates f on every vector in V,
or as an evaluation map (⋅)x ∈ V := L(V*, F), i.e. a linear function (⋅)x: V* --> F, which evaluates every functional in V* at x.
correct? ;)
Yeah :)
That's really fantastic . Thanks so much
I dont understand maybe ninety percent of what u say but i just like u so much i eatch every video!
We can go further!
press F to pay respects to HUMUNGOUS F
YESS I find these videos days before my linear algebra final
You keep saying things are different with infinite dimentional vector spaces. Can you give an example?
Thanks.
For example, in infinite dimensions, V* is not isomorphic to V
This is your funniest video 😊
I had so much fun making it 😂😂😂
YAY ITS YOU
V(ascot(balmoral(baseball(beanie(bearskin(beret(bicorne(bowler(chullo(cloche(cricket(sombrero(....(x))...) is a lot of dual hats (partially listed from wikipedia)
I LIKE U
Thanks ☺️
@@drpeyam hahaha no need to thank me. You are just naturally a likeable person. I like to say 'earth is lucky to have you'. It is especially true about you!
Awwwwwwww ❤️❤️❤️
@@drpeyam yay u repllied 💙
Voice is really annoying, am I the only one that thinks that?
For me it's fine.