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60-572: The Bakhta is 498 kilometres (309 mi) long, and the area of its basin is 35,500 square kilometres (13,700 sq mi). The lower reaches of the river are navigable. The Bakhta has its source in the Tunguska Plateau , part of the western side of the Central Siberian Plateau . It begins flowing northwestwards, then it bends about midway through its course and flows roughly southwestwards. The Bakhta flows in

120-407: A 2 , ..., a k are in F form a linear subspace called the span of S . The span of S is also the intersection of all linear subspaces containing S . In other words, it is the smallest (for the inclusion relation) linear subspace containing S . A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vectors is linearly independent if

180-400: A definite integral : The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes x and y the formula is: Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces ). For example, if the side surface of a cylinder (or any prism )

240-417: A corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m ), square centimetres (cm ), square millimetres (mm ), square kilometres (km ), square feet (ft ), square yards (yd ), square miles (mi ), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units. The SI unit of area

300-447: A difference w – z , and the line segments wz and 0( w − z ) are of the same length and direction. The segments are equipollent . The four-dimensional system H {\displaystyle \mathbb {H} } of quaternions was discovered by W.R. Hamilton in 1843. The term vector was introduced as v = x i + y j + z k representing a point in space. The quaternion difference p – q also produces

360-473: A finite number of elements, V is a finite-dimensional vector space . If U is a subspace of V , then dim U ≤ dim V . In the case where V is finite-dimensional, the equality of the dimensions implies U = V . If U 1 and U 2 are subspaces of V , then where U 1 + U 2 denotes the span of U 1 ∪ U 2 . Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps . Their theory

420-398: A linear map T  : V → W , the image T ( V ) of V , and the inverse image T ( 0 ) of 0 (called kernel or null space), are linear subspaces of W and V , respectively. Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums where v 1 , v 2 , ..., v k are in S , and a 1 ,

480-507: A matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants". Benjamin Peirce published his Linear Associative Algebra (1872), and his son Charles Sanders Peirce extended the work later. The telegraph required an explanatory system, and

540-404: A part of the basis of W is mapped bijectively on a part of the basis of V , and that the remaining basis elements of W , if any, are mapped to zero. Gaussian elimination is the basic algorithm for finding these elementary operations, and proving these results. A finite set of linear equations in a finite set of variables, for example, x 1 , x 2 , ..., x n , or x , y , ..., z

600-428: A rectangle with length l and width w , the formula for the area is: That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom . On

660-588: A remote mountainous area through a narrow valley surrounded by taiga until it leaves the plateau area and flows across the Yenisei plain. The Bakhta joins the right bank of the Yenisey at Bakhta village . The confluence is located roughly halfway between the mouths of the Podkamennaya Tunguska and Nizhnyaya Tunguska . The river freezes in mid-October and stays frozen until mid-May. A section of

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720-408: A segment equipollent to pq . Other hypercomplex number systems also used the idea of a linear space with a basis . Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group . The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote

780-556: A sphere was first obtained by Archimedes in his work On the Sphere and Cylinder . The formula is: where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus . Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices . Linear algebra

840-405: A vector by its inverse image under this isomorphism, that is by the coordinate vector ( a 1 , ..., a m ) or by the column matrix If W is another finite dimensional vector space (possibly the same), with a basis ( w 1 , ..., w n ) , a linear map f from W to V is well defined by its values on the basis elements, that is ( f ( w 1 ), ..., f ( w n )) . Thus, f

900-530: A vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra . The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. Until

960-457: Is a set V equipped with two binary operations . Elements of V are called vectors , and elements of F are called scalars . The first operation, vector addition , takes any two vectors v and w and outputs a third vector v + w . The second operation, scalar multiplication , takes any scalar a and any vector v and outputs a new vector a v . The axioms that addition and scalar multiplication must satisfy are

1020-428: Is a spanning set such that S ⊆ T , then there is a basis B such that S ⊆ B ⊆ T . Any two bases of a vector space V have the same cardinality , which is called the dimension of V ; this is the dimension theorem for vector spaces . Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension. If any basis of V (and therefore every basis) has

1080-447: Is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r , and the width is half the circumference of the circle, or π r . Thus, the total area of the circle is π r : Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of

1140-425: Is called a system of linear equations or a linear system . Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory has been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems. For example, let be a linear system. To such

1200-594: Is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry , including for defining basic objects such as lines , planes and rotations . Also, functional analysis , a branch of mathematical analysis , may be viewed as the application of linear algebra to function spaces . Linear algebra is also used in most sciences and fields of engineering , because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems , which cannot be modeled with linear algebra, it

1260-418: Is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone , the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature , it cannot be flattened out. The formula for the surface area of

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1320-462: Is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica , written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that

1380-437: Is often used for dealing with first-order approximations , using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point. The procedure (using counting rods) for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on

1440-431: Is related to the definition of determinants in linear algebra , and is a basic property of surfaces in differential geometry . In analysis , the area of a subset of the plane is defined using Lebesgue measure , though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through

1500-470: Is the square metre, which is considered an SI derived unit . Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m . This is equivalent to 6 million square millimetres. Other useful conversions are: In non-metric units,

1560-568: Is thus an essential part of linear algebra. Let V be a finite-dimensional vector space over a field F , and ( v 1 , v 2 , ..., v m ) be a basis of V (thus m is the dimension of V ). By definition of a basis, the map is a bijection from F , the set of the sequences of m elements of F , onto V . This is an isomorphism of vector spaces, if F is equipped of its standard structure of vector space, where vector addition and scalar multiplication are done component by component. This isomorphism allows representing

1620-509: Is used to refer to the region, as in a " polygonal area ". The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m ), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics ,

1680-415: Is well represented by the list of the corresponding column matrices. That is, if for j = 1, ..., n , then f is represented by the matrix with m rows and n columns. Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing

1740-460: The Cartesian coordinates ( x i , y i ) {\displaystyle (x_{i},y_{i})} ( i =0, 1, ..., n -1) of whose n vertices are known, the area is given by the surveyor's formula : where when i = n -1, then i +1 is expressed as modulus n and so refers to 0. The most basic area formula is the formula for the area of a rectangle . Given

1800-478: The hectare is still commonly used to measure land: Other uncommon metric units of area include the tetrad , the hectad , and the myriad . The acre is also commonly used to measure land areas, where An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns , such that: The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics . In South Asia (mainly Indians), although

1860-417: The surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. For a non-self-intersecting ( simple ) polygon,

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1920-429: The unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number . There are several well-known formulas for the areas of simple shapes such as triangles , rectangles , and circles . Using these formulas, the area of any polygon can be found by dividing the polygon into triangles . For shapes with curved boundary, calculus is usually required to compute

1980-527: The 1873 publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression. Linear algebra is flat differential geometry and serves in tangent spaces to manifolds . Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations , and much of the history of linear algebra is the history of Lorentz transformations . The first modern and more precise definition of

2040-406: The 19th century, linear algebra was introduced through systems of linear equations and matrices . In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic , more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract. A vector space over a field F (often the field of the real numbers )

2100-556: The 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , but did not identify the constant of proportionality . Eudoxus of Cnidus , also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared. Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used

2160-819: The Mathematical Art . Its use is illustrated in eighteen problems, with two to five equations. Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry . In fact, in this new geometry, now called Cartesian geometry , lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. The first systematic methods for solving linear systems used determinants and were first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule . Later, Gauss further described

2220-439: The amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have the same area (as in squaring the circle ); by synecdoche , "area" sometimes

2280-470: The area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula , for the area of any quadrilateral. The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of

2340-507: The area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry . In 499 Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , expressed the area of a triangle as one-half the base times the height in the Aryabhatiya . In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula , for

2400-581: The area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus . For a solid shape such as a sphere , cone, or cylinder, the area of its boundary surface is called the surface area . Formulas for the surface areas of simple shapes were computed by the ancient Greeks , but computing the surface area of a more complicated shape usually requires multivariable calculus . Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area

2460-417: The areas of the approximate parallelograms is exactly π r , which is the area of the circle. This argument is actually a simple application of the ideas of calculus . In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus . Using modern methods, the area of a circle can be computed using

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2520-417: The conversion between two square units is the square of the conversion between the corresponding length units. the relationship between square feet and square inches is where 144 = 12 = 12 × 12. Similarly: In addition, conversion factors include: There are several other common units for area. The are was the original unit of area in the metric system , with: Though the are has fallen out of use,

2580-523: The countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used. Some traditional South Asian units that have fixed value: In

2640-549: The following. (In the list below, u , v and w are arbitrary elements of V , and a and b are arbitrary scalars in the field F .) The first four axioms mean that V is an abelian group under addition. An element of a specific vector space may have various nature; for example, it could be a sequence , a function , a polynomial or a matrix . Linear algebra is concerned with those properties of such objects that are common to all vector spaces. Linear maps are mappings between vector spaces that preserve

2700-417: The induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces . More precisely, a linear subspace of a vector space V over a field F is a subset W of V such that u + v and a u are in W , for every u , v in W , and every a in F . (These conditions suffice for implying that W is a vector space.) For example, given

2760-406: The left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of

2820-525: The lower course of the river, including its confluence with the Yenisei are located in the Central Siberia Nature Reserve . Area Area is the measure of a region 's size on a surface . The area of a plane region or plane area refers to the area of a shape or planar lamina , while surface area refers to the area of an open surface or the boundary of a three-dimensional object . Area can be understood as

2880-492: The method of elimination, which was initially listed as an advancement in geodesy . In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix , which is Latin for womb . Linear algebra grew with ideas noted in the complex plane . For instance, two numbers w and z in C {\displaystyle \mathbb {C} } have

2940-407: The only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a i . A set of vectors that spans a vector space is called a spanning set or generating set . If a spanning set S is linearly dependent (that is not linearly independent), then some element w of S is in the span of the other elements of S , and the span would remain

3000-431: The other by elementary row and column operations . For a matrix representing a linear map from W to V , the row operations correspond to change of bases in V and the column operations correspond to change of bases in W . Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from W to V , there are bases such that

3060-432: The other hand, if geometry is developed before arithmetic , this formula can be used to define multiplication of real numbers . Most other simple formulas for area follow from the method of dissection . This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle , as shown in figure to

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3120-421: The parallelogram: Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons . The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk ) is based on a similar method. Given a circle of radius r , it is possible to partition the circle into sectors , as shown in the figure to the right. Each sector

3180-424: The result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing exactly the same concepts. Two matrices that encode the same linear transformation in different bases are called similar . It can be proved that two matrices are similar if and only if one can transform one into

3240-419: The same if one were to remove w from S . One may continue to remove elements of S until getting a linearly independent spanning set . Such a linearly independent set that spans a vector space V is called a basis of V . The importance of bases lies in the fact that they are simultaneously minimal generating sets and maximal independent sets. More precisely, if S is a linearly independent set, and T

3300-416: The same vector space, a linear map T  : V → V is also known as a linear operator on V . A bijective linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an isomorphism . Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in

3360-524: The sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm . The study of those subsets of vector spaces that are in themselves vector spaces under

3420-410: The tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle . (The circumference is 2 π r , and the area of a triangle is half the base times the height, yielding the area π r for the disk.) Archimedes approximated

3480-511: The use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. An approach to defining what is meant by "area" is through axioms . "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties: It can be proved that such an area function actually exists. Every unit of length has

3540-419: The value of π (and hence the area of a unit-radius circle) with his doubling method , in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon , then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons ). Heron of Alexandria found what

3600-432: The vector-space structure. Given two vector spaces V and W over a field F , a linear map (also called, in some contexts, linear transformation or linear mapping) is a map that is compatible with addition and scalar multiplication, that is for any vectors u , v in V and scalar a in F . This implies that for any vectors u , v in V and scalars a , b in F , one has When V = W are

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