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
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
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
In this video, we solve a classical dual space exercise: Given a set F of linear functionals, find a basis B of V such that F is the dual basis of B. This procedure is very important in applications, an in fact in another video, we'll see a neat application of this idea to numerical integr
From playlist Dual Spaces
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
35 - Properties of bases (continued)
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
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
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
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
What is General Relativity? Lesson 47: The double dual of the Riemann tensor
What is General Relativity? Lesson 47: The double dual of the Riemann Tensor The double dual of the Riemann tensor is an obscure object but it gives us a chance to quickly review 2-forms. Please consider supporting this channel via Patreon: https://www.patreon.com/XYLYXYLYX and discuss
From playlist What is General Relativity?
[Lesson 2] QED Prerequisites Dirac Formalism Part 2
In this second lesson on the Dirac formalism we make the connection between bras and kets by defining the "dual correspondence" via an inner product on V. Please consider supporting this channel on Patreon: https://www.patreon.com/XYLYXYLYX The software I usually use to produce the lect
From playlist QED- Prerequisite Topics
[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
Linear functionals, dual spaces, dual bases, and dual maps.
From playlist Linear Algebra Done Right
[Lesson 3] QED Prerequisites Dirac Formalism Part 3
This lesson is about the Dirac formalism's approach to linear operators. These operators will be the core of the theory of quantum mechanics, and the Dirac formalism is a very tight way of understanding them. [reposted to fix small error in title screen] Please consider supporting this c
From playlist QED- Prerequisite Topics
Maxwell's Equations via Differential Forms Part 2
In this lesson we review the Hodge star operator and the concept of the Hodge dual of a vector. We present and demonstrate a specific formula that calculates the Hodge dual of any k-form. The purpose of this is to set ourselves up to cast Maxwell's Equations in the language of differential
From playlist QED- Prerequisite Topics
Solving a multi step equation using distributive property
👉 Learn how to solve multi-step equations with parenthesis and variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To
From playlist How to Solve Multi Step Equations with Parenthesis on Both Sides