Dual space

Dual Space

Dual spaces and linear functionals In this video, I introduce the concept of a dual space, which is the analog of a "shadow world" version, but for vector spaces. I also give some examples of linear and non-linear functionals. This seems like an innocent topic, but it has a huge number of

From playlist Dual Spaces

Dual basis

Dual basis definition and proof that it's a basis In this video, given a basis beta of a vector space V, I define the dual basis beta* of V*, and show that it's indeed a basis. We'll see many more applications of this concept later on, but this video already shows that it's straightforwar

From playlist Dual Spaces

Dual Basis Example

In this video, I show how to explicitly calculate dual bases. More specifically, I find the dual basis corresponding to the basis (2,1) and (3,1) of R^2. Hopefully this will give you a better idea of how dual bases work. Subscribe to my channel: https://www.youtube.com/c/drpeyam What is

From playlist Dual Spaces

Double Dual

Definition of V** (double dual) and an amazing miracle Dual Space Definition: https://youtu.be/OGO3HGlOQO4 Dual Spaces Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCs0fJDQnXgeuyFR8iQDwLV Subscribe to my channel: https://www.youtube.com/c/drpeyam

From playlist Dual Spaces

Duality Argument

In this video, I present a very classical example of a duality argument: Namely, I show that T^T is one-to-one if and only if T is onto and use that to show that T is one-to-one if and only if T^T is onto. This illustrates the beautiful interplay between a vector space and its dual space,

From playlist Dual Spaces

Dirac Delta

Dirac Delta Definition In this video, I define the Dirac Delta functional, and show that it is strictly speaking not a function. Along the way, I show that for infinite dimensions, a vector space is not necessarily isomorphic to its dual space. Enjoy! Check out my dual space playlist: ht

From playlist Dual Spaces

What the V*****

In this fun video, I determine explicitly what V***** (the quintuple dual) looks like. Enjoy! What is a Dual Space: https://youtu.be/OGO3HGlOQO4 Double Dual: https://youtu.be/3ntR99gyKnQ Check out my Dual Spaces Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCs0fJDQnXgeuyF

From playlist Dual Spaces

Every basis is a dual basis

In this video, I show a very neat result about dual spaces: Namely, any basis of V* is automatically a dual basis of some basis of V. Even though this result is very interesting, it's the proof that makes this very exciting, by simply using the fact that V and V** are 'very' isomorphic. En

From playlist Dual Spaces

Duality in Linear Algebra: Dual Spaces, Dual Maps, and All That

An exploration of duality in linear algebra, including dual spaces, dual maps, and dual bases, with connections to linear and bilinear forms, adjoints in real and complex inner product spaces, covariance and contravariance, and matrix rank. More videos on linear algebra: https://youtube.c

From playlist Summer of Math Exposition Youtube Videos

What is a Tensor 4: Cartesian Products

What is a Tensor 4: Cartesian Products

From playlist What is a Tensor?

QED Prerequisites Geometric Algebra 14: The Pseudoscalar

In this lesson we introduce the basis element of the grade 4 part of the spacetime algebra: the pseudoscalar. ERRATA: At about 6:00 I do a demonstration and slipped into the (-1,1,1,1) metric convention for a moment when I said (gamma_0)^2 = -1 …. An easy mistake to make! The result is st

From playlist QED- Prerequisite Topics

Dual Bases and Dual Maps

Linear functionals, dual spaces, dual bases, and dual maps.

From playlist Linear Algebra Done Right

What is a Tensor? Lesson 36: Other Notions of Duality

What is a Tensor? Lesson 36: Other Notions of Duality

From playlist What is a Tensor?

Rings 12 Duality and injective modules

This lecture is part of an online course on rings and modules. We descibe some notions of duality for modules generalizing the dual of a vector space. We first discuss duality for free and projective modules, which is very siilar to the vector space case. Then we discuss duality for finit

From playlist Rings and modules

What is a Tensor 6: Tensor Product Spaces

What is a Tensor 6: Tensor Product Spaces There is an error at 15:00 which is annotated but annotations can not be seen on mobile devices. It is a somewhat obvious error! Can you spot it? :)

From playlist What is a Tensor?

[Lesson 1] QED Prerequisites Dirac Formalism Part I (redux)

(Editorial repair made in this version) This lecture is the first in a series of topics related to QED prerequisite material. I will be selecting some topics that students are often not clear about when arriving at QED. These topics cover a wide variety of material in elementary quantum m

From playlist QED- Prerequisite Topics

What is a Tensor? Lesson 23: Operations on p-forms. The Exterior Algebra.

What is a Tensor? Lesson 23: Operations on p-forms. The Exterior Algebra.

From playlist What is a Tensor?

Transpose Definition

Definition of the transpose Have you ever wondered where the transpose comes from? In this video, I show that the transpose arises naturally in the setting of dual spaces. This should also illustrate why dual spaces are so important. Enjoy! Transpose Example (Sequel): https://youtu.be/x2

From playlist Dual Spaces

What is a Tensor? Lesson 35: Elementary Hodge Dual Calculations

What is a Tensor? Lesson 35: Elementary Hodge Dual Calculations

From playlist What is a Tensor?