A report on MathematicsGeometry and Vector space

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
An illustration of Desargues' theorem, a result in Euclidean and projective geometry
Addition of functions: The sum of the sine and the exponential function is
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
A European and an Arab practicing geometry in the 15th century
A typical matrix
The quadratic formula expresses concisely the solutions of all quadratic equations
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
Commutative diagram depicting the universal property of the tensor product.
Rubik's cube: the study of its possible moves is a concrete application of group theory
An illustration of Euclid's parallel postulate
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
An affine plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
A page from al-Khwārizmī's Algebra
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
A tiling of the hyperbolic plane
Leonhard Euler created and popularized much of the mathematical notation used today.
Differential geometry uses tools from calculus to study problems involving curvature.
Carl Friedrich Gauss, known as the prince of mathematicians
A thickening of the trefoil knot
The front side of the Fields Medal
Quintic Calabi–Yau threefold
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

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
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

5 related topics with Alpha

Overall

A point in three-dimensional Euclidean space can be located by three coordinates.

Euclidean space

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A point in three-dimensional Euclidean space can be located by three coordinates.
Positive and negative angles on the oriented plane
3-dimensional skew coordinates

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

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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.

In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Linear algebra

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In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

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.

In affine geometry, one uses Playfair's axiom to find the line through C1 and parallel to B1B2, and to find the line through B2 and parallel to B1C1: their intersection C2 is the result of the indicated translation.

Affine geometry

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In affine geometry, one uses Playfair's axiom to find the line through C1 and parallel to B1B2, and to find the line through B2 and parallel to B1C1: their intersection C2 is the result of the indicated translation.
Pappus's law: if the red lines are parallel and the blue lines are parallel, then the dotted black lines must be parallel.

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.

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.

Algebra

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Quadratic formula.svg expresses the solution of the equation

Quadratic formula.svg expresses the solution of the equation

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.
A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
Linear algebra lecture at the Aalto University
Algebraic expression notation:
 1 – power (exponent)
 2 – coefficient
 3 – term
 4 – operator
 5 – constant term
 x y c – variables/constants
The graph of a polynomial function of degree 3

Algebra is one of the broad areas of mathematics.

Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example).

At the beginning of the 20th century, algebra evolved again by considering operations that act not only on numbers but also on elements of so-called mathematical structures such as groups, fields and vector spaces.