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Euler introduced the term affine (Latin affinis, "related") in 1748 in his book "Introductio in analysin infinitorum."

Felix Klein's Erlangen program recognized affine geometry as a generalization of Euclidean geometry.

Q: Who invented affine space in linear algebra?

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An affine combination is a linear combination of vectors in Euclidian space in which the coefficients add up to one.

View all Sir William Rowan Hamilton invented the linear equation in 1843.

The history of modern linear algebra dates back to the early 1840's. In 1843, William Rowan Hamilton introduced quaternions, which describe mechanics in three-dimensional space. In 1844, Hermann Grassmann published his book Die lineale Ausdehnungslehre (see References). Arthur Cayley introduced matrices, one of the most fundamental linear algebraic ideas, in 1857. Despite these early developments, linear algebra has been developed primarily in the twentieth century.

dual space W* of W can naturally identified with linear functionals

Algebra, shape space and measure

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An affine combination is a linear combination of vectors in Euclidian space in which the coefficients add up to one.

An affine space is a vector space with no origin.

An affine group is the group of all affine transformations of a finite-dimensional vector space.

An affine variety is a set of points in n-dimensional space which satisfy a set of equations which have a polynomial of n variables on one side and a zero on the other side.

View all Sir William Rowan Hamilton invented the linear equation in 1843.

Linear algebra is used to analyze systems of linear equations. Oftentimes, these systems of linear equations are very large, making up many, many equations and are many dimensions large. While students should never have to expect with anything larger than 5 dimensions (R5 space), in real life, you might be dealing with problems which have 20 dimensions to them (such as in economics, where there are many variables). Linear algebra answers many questions. Some of these questions are: How many free variables do I have in a system of equations? What are the solutions to a system of equations? If there are an infinite number of solutions, how many dimensions do the solutions span? What is the kernel space or null space of a system of equations (under what conditions can a non-trivial solution to the system be zero?) Linear algebra is also immensely valuable when continuing into more advanced math topics, as you reuse many of the basic principals, such as subspaces, basis, eigenvalues and not to mention a greatly increased ability to understand a system of equations.

In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.

The history of modern linear algebra dates back to the early 1840's. In 1843, William Rowan Hamilton introduced quaternions, which describe mechanics in three-dimensional space. In 1844, Hermann Grassmann published his book Die lineale Ausdehnungslehre (see References). Arthur Cayley introduced matrices, one of the most fundamental linear algebraic ideas, in 1857. Despite these early developments, linear algebra has been developed primarily in the twentieth century.

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dual space W* of W can naturally identified with linear functionals

It allowed points in space to be described algebraically. This allowed lines and curves to be described using algebra. Bringing together algebra and geometry meant that tools that mathematicians had developed for solving algebraic problems could be applied to problems in geometry and tools from geometry could be applied to algebra.

YesBy defining ab=0 (zero product) every Banach space become a Banach algebra.