A report onMathematics, Geometry and Vector space

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

Geometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry

In mathematics, physics, and engineering, 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.

- Vector space

Finite-dimensional vector spaces occur naturally in geometry and related areas.

- Vector space

vector spaces, whose study is essentially the same as linear algebra;

- Mathematics

A similar and closely related form of duality exists between a vector space and its dual space.

- Geometry

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane (dimension two).

Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition.

Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

On a finite-dimensional vector space this topology is the same for all norms.

Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as:

and their representations in vector spaces and through matrices.

For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations.

Affine geometry

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting" ) the metric notions of distance and angle.

In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.

In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry.