In mathematics, the term linear is used in two distinct senses for two different properties:
41-470: Not to be confused with Linearity . [REDACTED] Look up linage , lineage , or lineages in Wiktionary, the free dictionary. Lineage may refer to: Science [ edit ] Lineage (anthropology) , a group that can demonstrate its common descent from an apical ancestor or a direct line of descent from an ancestor Lineage (evolution) ,
82-519: A 0 = 1 {\displaystyle a_{0}=1} , the above function is considered affine in linear algebra (i.e. not linear). A Boolean function is linear if one of the following holds for the function's truth table : Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference. Negation , Logical biconditional , exclusive or , tautology , and contradiction are linear functions. In physics , linearity
123-541: A plane is a flat two- dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below: The Schläfli symbol { n } {\displaystyle \{n\}} represents
164-419: A regular n -gon . The regular monogon (or henagon) {1} and regular digon {2} can be considered degenerate regular polygons and exist nondegenerately in non-Euclidean spaces like a 2-sphere , 2-torus , or right circular cylinder . There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share
205-564: A certain operating region—for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value. For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes: There are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well
246-491: A linear polynomial in the variables X , {\displaystyle X,} Y {\displaystyle Y} and Z {\displaystyle Z} is a X + b Y + c Z + d . {\displaystyle aX+bY+cZ+d.} Linearity of a mapping is closely related to proportionality . Examples in physics include the linear relationship of voltage and current in an electrical conductor ( Ohm's law ), and
287-533: A region D in R of a function f ( x , y ) , {\displaystyle f(x,y),} and is usually written as: The fundamental theorem of line integrals says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let φ : U ⊆ R 2 → R {\displaystyle \varphi :U\subseteq \mathbb {R} ^{2}\to \mathbb {R} } . Then with p , q
328-826: A series of short films based on the Assassin's Creed II video game Television [ edit ] "Lineage" ( Angel ) , a 2003 episode of the television series Angel "Lineage" ( Smallville ) , a 2002 episode of television series Smallville "Lineage" ( Star Trek: Voyager ) , a 2001 episode of the television series Star Trek: Voyager "Lineage", a 2012 episode of the television series Revenge (season 2) Other fields [ edit ] Embraer Lineage , business jet Lineage (Buddhism) , line of transmission of Buddhist teachers Zen lineage charts , lines of transmission of Zen teachers Line of transmission of martial arts teachers and/or founders LineageOS , an open-source operating system Topics referred to by
369-560: A temporal sequence of individuals, populations or species which represents a continuous line of descent Lineage (genetic) Lineage markers Data lineage Gaming [ edit ] Lineage (series) , a medieval fantasy massively multiplayer online role-playing game franchise Lineage (video game) , the original 1998 game Lineage II , a 2003 prequel to Lineage Lineage III , an upcoming sequel to Lineage II Lineage W , 3D mobile version of Lineage aiming for global service Assassin's Creed: Lineage ,
410-468: A vector A by itself is which gives the formula for the Euclidean length of the vector. In a rectangular coordinate system, the gradient is given by For some scalar field f : U ⊆ R → R , the line integral along a piecewise smooth curve C ⊂ U is defined as where r : [a, b] → C is an arbitrary bijective parametrization of the curve C such that r ( a ) and r ( b ) give
451-499: Is a geometric space in which two real numbers are required to determine the position of each point . It is an affine space , which includes in particular the concept of parallel lines . It has also metrical properties induced by a distance , which allows to define circles , and angle measurement . A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane . The set R 2 {\displaystyle \mathbb {R} ^{2}} of
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#1732858646449492-400: Is a high fidelity audio amplifier , which must amplify a signal without changing its waveform. Others are linear filters , and linear amplifiers in general. In most scientific and technological , as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within
533-1201: Is a function that satisfies the two properties: These properties are known as the superposition principle . In this definition, x is not necessarily a real number , but can in general be an element of any vector space . A more special definition of linear function , not coinciding with the definition of linear map, is used in elementary mathematics (see below). Additivity alone implies homogeneity for rational α, since f ( x + x ) = f ( x ) + f ( x ) {\displaystyle f(x+x)=f(x)+f(x)} implies f ( n x ) = n f ( x ) {\displaystyle f(nx)=nf(x)} for any natural number n by mathematical induction , and then n f ( x ) = f ( n x ) = f ( m n m x ) = m f ( n m x ) {\displaystyle nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)} implies f ( n m x ) = n m f ( x ) {\displaystyle f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)} . The density of
574-408: Is a one-dimensional manifold . In a Euclidean plane, it has the length 2π r and the area of its interior is where r {\displaystyle r} is the radius. There are an infinitude of other curved shapes in two dimensions, notably including the conic sections : the ellipse , the parabola , and the hyperbola . Another mathematical way of viewing two-dimensional space
615-464: Is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation . Linearity of a homogenous differential equation means that if two functions f and g are solutions of the equation, then any linear combination af + bg is, too. In instrumentation, linearity means that a given change in an input variable gives the same change in
656-437: Is characterized as being the unique contractible 2-manifold . Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not simply connected . In graph theory , a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such
697-430: Is defined as: A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by ‖ A ‖ {\displaystyle \|\mathbf {A} \|} . In this viewpoint, the dot product of two Euclidean vectors A and B is defined by where θ is the angle between A and B . The dot product of
738-401: Is different from Wikidata All article disambiguation pages All disambiguation pages Linearity An example of a linear function is the function defined by f ( x ) = ( a x , b x ) {\displaystyle f(x)=(ax,bx)} that maps the real line to a line in the Euclidean plane R that passes through the origin. An example of
779-440: Is found in linear algebra , where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors . The dot product of two vectors A = [ A 1 , A 2 ] and B = [ B 1 , B 2 ]
820-424: Is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the other axis. Another widely used coordinate system is the polar coordinate system , which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray. In Euclidean geometry ,
861-416: Is less than two. The use of the term for polynomials stems from the fact that the graph of a polynomial in one variable is a straight line . In the term " linear equation ", the word refers to the linearity of the polynomials involved. Because a function such as f ( x ) = a x + b {\displaystyle f(x)=ax+b} is defined by a linear polynomial in its argument, it
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#1732858646449902-419: Is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context. The word linear comes from Latin linearis , "pertaining to or resembling a line". In mathematics, a linear map or linear function f ( x )
943-417: Is the branch of mathematics concerned with systems of linear equations. In Boolean algebra , a linear function is a function f {\displaystyle f} for which there exist a 0 , a 1 , … , a n ∈ { 0 , 1 } {\displaystyle a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}} such that Note that if
984-427: The device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale , or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in
1025-425: The discovery. Both authors used a single ( abscissa ) axis in their treatments, with the lengths of ordinates measured along lines not-necessarily-perpendicular to that axis. The concept of using a pair of fixed axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify
1066-462: The endpoints of C and a < b {\displaystyle a<b} . For a vector field F : U ⊆ R → R , the line integral along a piecewise smooth curve C ⊂ U , in the direction of r , is defined as where · is the dot product and r : [a, b] → C is a bijective parametrization of the curve C such that r ( a ) and r ( b ) give the endpoints of C . A double integral refers to an integral within
1107-399: The endpoints of the curve γ. Let C be a positively oriented , piecewise smooth , simple closed curve in a plane , and let D be the region bounded by C . If L and M are functions of ( x , y ) defined on an open region containing D and have continuous partial derivatives there, then where the path of integration along C is counterclockwise . In topology , the plane
1148-424: The equation up into smaller pieces, solving each of those pieces, and summing the solutions. In a different usage to the above definition, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a straight line. Over the reals, a simple example of a linear equation is given by: where m is often called the slope or gradient , and b the y-intercept , which gives
1189-608: The ideas contained in Descartes' work. Later, the plane was thought of as a field , where any two points could be multiplied and, except for 0, divided. This was known as the complex plane . The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot
1230-400: The linear operating region of a device, for example a transistor , is where an output dependent variable (such as the transistor collector current ) is directly proportional to an input dependent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment
1271-525: The manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics. Euclidean plane In mathematics , a Euclidean plane is a Euclidean space of dimension two , denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E} ^{2}} . It
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1312-496: The ordered pairs of real numbers (the real coordinate plane ), equipped with the dot product , is often called the Euclidean plane or standard Euclidean plane , since every Euclidean plane is isomorphic to it. Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas , parallelism,
1353-433: The output of the measurement apparatus: this is highly desirable in scientific work. In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain absolute threshold number of photons. Linear motion traces a straight line trajectory. In electronics ,
1394-490: The point of intersection between the graph of the function and the y -axis. Note that this usage of the term linear is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if the constant term – b in the example – equals 0. If b ≠ 0 , the function is called an affine function (see in greater generality affine transformation ). Linear algebra
1435-427: The positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin . They are usually labeled x and y . Relative to these axes, the position of any point in two-dimensional space
1476-545: The rational numbers in the reals implies that any additive continuous function is homogeneous for any real number α, and is therefore linear. The concept of linearity can be extended to linear operators . Important examples of linear operators include the derivative considered as a differential operator , and other operators constructed from it, such as del and the Laplacian . When a differential equation can be expressed in linear form, it can generally be solved by breaking
1517-401: The relationship of mass and weight . By contrast, more complicated relationships, such as between velocity and kinetic energy , are nonlinear . Generalized for functions in more than one dimension , linearity means the property of a function of being compatible with addition and scaling , also known as the superposition principle . Linearity of a polynomial means that its degree
1558-542: The same unit of length . Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat , although Fermat also worked in three dimensions, and did not publish
1599-404: The same vertex arrangements of the convex regular polygons. In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols { n / m } for all m such that m < n /2 (strictly speaking { n / m } = { n /( n − m )}) and m and n are coprime . The hypersphere in 2 dimensions is a circle , sometimes called a 1-sphere ( S ) because it
1640-411: The same term [REDACTED] This disambiguation page lists articles associated with the title Lineage . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Lineage&oldid=1171795938 " Category : Disambiguation pages Hidden categories: Short description
1681-432: The sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called Cartesian coordinate system , a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates , which are the signed distances from the point to two fixed perpendicular directed lines, measured in