Definition of linear map. Algebraic properties of linear maps.
From playlist Linear Algebra Done Right
Invertibility and Isomorphic Vector Spaces
The dimension of L(V, W). Linear maps act like matrix multiplication. Injectivity is equivalent to surjectivity in finite dimensions.
From playlist Linear Algebra Done Right
A map T between vector spaces which satisfies: * the addition condition T(x+y) = T(x) + T(y) and * the scalar multiplication condition T(lambda x) = lambda T(x) is called a "linear map". In the first part of this video we see how to show a map is linear; in the second part we see how t
From playlist Mathematics 1B (Algebra)
Linear Algebra: Here are a few problems on linear maps. Part 1: Are the following maps L:R^3 to R^3 linear? (a) L(x, y, z) = (x+1, x-y-2, y-z), (b) L(x, y, z) = (x + 2y, x-y-2z, 0). Part 2: Suppose L:R^3 to R^2 is linear and defined on the standard basis by L(e1) = (1, 2), L(e2) =
From playlist MathDoctorBob: Linear Algebra I: From Linear Equations to Eigenspaces | CosmoLearning.org Mathematics
André JOYAL - 3/4 A crash course in topos theory : the big picture
I will sketch an overall picture of topos theory and of the theory of locales. It includes the notion of sheaf on a site, the notion of forcing topology, of geometric morphism and Giraud's theorem. A useful principle is that a topos is a commutative ring-like object. Every topos is a quoti
From playlist Topos à l'IHES
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
André JOYAL - 2/4 A crash course in topos theory : the big picture
I will sketch an overall picture of topos theory and of the theory of locales. It includes the notion of sheaf on a site, the notion of forcing topology, of geometric morphism and Giraud's theorem. A useful principle is that a topos is a commutative ring-like object. Every topos is a quoti
From playlist Topos à l'IHES
André JOYAL - 4/4 A crash course in topos theory : the big picture
I will sketch an overall picture of topos theory and of the theory of locales. It includes the notion of sheaf on a site, the notion of forcing topology, of geometric morphism and Giraud's theorem. A useful principle is that a topos is a commutative ring-like object. Every topos is a quoti
From playlist Topos à l'IHES
MAST30026 Lecture 19: Duality and Hilbert space
I began by proving the universal property of the completion of a normed space. I then discussed characterisations of finite-dimensionality for vector spaces, introduced the continuous linear dual for normed spaces and the operator norm, and stated the duality theorem or L^p spaces which sa
From playlist MAST30026 Metric and Hilbert spaces
SEPARATION BUT MATHEMATICALLY: What Types of Mathematical Topologies are there? | Nathan Dalaklis
The title of this video is a bit convoluted. What do you mean by "Separation but Mathematically"? Well, in this video I'll be giving a (very diluted) answer to the question "What types of mathematical topologies are there?" by introducing the separation axioms in topology. The separation
From playlist The New CHALKboard
Johnathan Bush (11/5/21): Maps of Čech and Vietoris–Rips complexes into euclidean spaces
We say a continuous injective map from a topological space to k-dimensional euclidean space is simplex-preserving if the image of each set of at most k+1 distinct points is affinely independent. We will describe how simplex-preserving maps can be useful in the study of Čech and Vietoris–Ri
From playlist Vietoris-Rips Seminar
Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs
❤️ Become The AI Epiphany Patreon ❤️ https://www.patreon.com/theaiepiphany 👨👩👧👦 Join our Discord community 👨👩👧👦 https://discord.gg/peBrCpheKE In this video I cover "Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs" paper. The paper takes id
From playlist Graph Neural Nets
Index Theory, survey - Stephan Stolz [2018]
TaG survey series These are short series of lectures focusing on a topic in geometry and topology. May_8_2018 Stephan Stolz - Index Theory https://www3.nd.edu/~math/rtg/tag.html (audio fixed)
From playlist Mathematics
Marie Kerjean: Differential linear logic extended to differential operators
HYBRID EVENT Recorded during the meeting Linear Logic Winter School" the January 28, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual
From playlist Logic and Foundations
Stefan Schwede: Equivariant stable homotopy - Lecture 1
I will use the orthogonal spectrum model to introduce the tensor triangulated category of genuine G-spectra, for compact Lie groups G. I will explain structural properties such as the smash product of G-spectra, and functors relating the categories for varying G (fixed points, geometric fi
From playlist Summer School: Spectral methods in algebra, geometry, and topology
Florian Frick (6/4/21): Rips complexes, projective codes, and zeros of odd maps
We will discuss a relation between the topology of Rips complexes (or their metric versions), the size of codes in projective spaces, and structural results for the zero set of odd maps from spheres to Euclidean space. On the one hand, this provides a new topological approach to problems i
From playlist Vietoris-Rips Seminar
Joshua Mike (6/15/20): TALLEM: Topological Assembly of Locally Linear Euclidean Models
Title: TALLEM: Topological Assembly of Locally Linear Euclidean Models Abstract: We present a new topological data analysis tool for nonlinear dimensionality reduction. This method, dubbed TALLEM, assembles a collection of local Euclidean coordinates, and leverages ideas from the theory o
From playlist ATMCS/AATRN 2020
Topological Filters via Sheaves [Georg Essl]
This tutorial describes the basic construction of topological filters via sheaves. As a concrete example it shows how linear time-invariant filters can be realized over a simplicial line complex. It concludes by demonstrating a real-life rendering of sound of a filter-based plucked string
From playlist Tutorial-a-thon 2021 Fall
After our introduction to matrices and vectors and our first deeper dive into matrices, it is time for us to start the deeper dive into vectors. Vector spaces can be vectors, matrices, and even function. In this video I talk about vector spaces, subspaces, and the porperties of vector sp
From playlist Introducing linear algebra
