Lecture

Differential of a Morphism

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Description

This lecture covers the concept of the differential of a morphism, which is a generalization of the differential known from analysis. It explains how to linearly approximate a map by using tangent spaces, defining the differential as a linear map between tangent spaces. The lecture also explores the properties of differentials, such as their behavior under composition and with constant maps. Additionally, it demonstrates the use of dual numbers to calculate differentials and provides examples of calculating differentials for matrix multiplication, emphasizing the importance of differentials in algebraic treatments.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Geometry: Algebraic geometry

Related concepts (32)

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Algebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.

Matrix multiplication

In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

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