Transformation (function) | Abstract algebra | Linear operators | Functions and mappings
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a linear isomorphism. In the case where , a linear map is called a (linear) endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that and are real vector spaces (not necessarily with ), or it can be used to emphasize that is a function space, which is a common convention in functional analysis. Sometimes the term linear function has the same meaning as linear map, while in analysis it does not. A linear map from V to W always maps the origin of V to the origin of W. Moreover, it maps linear subspaces in V onto linear subspaces in W (possibly of a lower dimension); for example, it maps a plane through the origin in V to either a plane through the origin in W, a line through the origin in W, or just the origin in W. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of category theory, linear maps are the morphisms of vector spaces. (Wikipedia).
In this lecture, we discuss the definition of a linear map, relating it to the definition of a vector space. We also give an elementary example, illustrating how to verify that a map is linear or not linear. Video Notes: https://theorembmath.files.wordpress.com/2020/02/linear-maps-definit
From playlist Linear Algebra
What is the slope of a linear equation
👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu
From playlist ⚡️Graph Linear Equations | Learn About
👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu
From playlist ⚡️Graph Linear Equations | Learn About
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
Definition of linear map. Algebraic properties of linear maps.
From playlist Linear Algebra Done Right
👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu
From playlist ⚡️Graph Linear Equations | Learn About
👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu
From playlist ⚡️Graph Linear Equations | Learn About
Showing something is a linear transformation Check out my Linear Equations playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmD_u31hoZ1D335sSKMvVQ90 Subscribe to my channel: https://www.youtube.com/channel/UCoOjTxz-u5zU0W38zMkQIFw
From playlist Linear Transformations
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
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
56 - Operations on linear maps
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
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
Lek-Heng Lim: "What is a tensor? (Part 2/2)"
Watch part 1/2 here: https://youtu.be/MkYEh0UJKcE Tensor Methods and Emerging Applications to the Physical and Data Sciences Tutorials 2021 "What is a tensor? (Part 2/2)" Lek-Heng Lim - University of Chicago, Statistics Abstract: We discuss the three best-known definitions of a tensor:
From playlist Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021
Linear functionals, dual spaces, dual bases, and dual maps.
From playlist Linear Algebra Done Right
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
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
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)
Part III: Linear Algebra, Lec 4: Linear Transformations
Part III: Linear Algebra, Lecture 4: Linear Transformations Instructor: Herbert Gross View the complete course: http://ocw.mit.edu/RES18-008F11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT Calculus Revisited: Calculus of Complex Variables