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| http://dbpedia.org/ontology/abstract | In mathematics, a linear fractional transf … In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form which has an inverse. The precise definition depends on the nature of a, b, c, d, and z. In other words, a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator are linear. In the most basic setting, a, b, c, d, and z are complex numbers (in which case the transformation is also called a Möbius transformation), or more generally elements of a field. The invertibility condition is then ad – bc ≠ 0. Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. When a, b, c, d are integer (or, more generally, belong to an integral domain), z is supposed to be a rational number (or to belong to the field of fractions of the integral domain. In this case, the invertibility condition is that ad – bc must be a unit of the domain (that is 1 or −1 in the case of integers). In the most general setting, the a, b, c, d and z are square matrices, or, more generally, elements of a ring. An example of such linear fractional transformation is the Cayley transform, which was originally defined on the 3 x 3 real matrix ring. Linear fractional transformations are widely used in various areas of mathematics and its applications to engineering, such as classical geometry, number theory (they are used, for example, in Wiles's proof of Fermat's Last Theorem), group theory, control theory.st Theorem), group theory, control theory. , Lineární lomená funkce je funkce, kterou lze zapsat ve tvaru . , Дро́бно-лине́йное преобразова́ние или дро́бно-лине́йное отображе́ние — это отображение комплексного пространства на себя, которое осуществляется дробно-линейными функциями. , 数学の特に複素解析における一次分数変換（いちじぶんすうへんかん、英: linear … 数学の特に複素解析における一次分数変換（いちじぶんすうへんかん、英: linear fractional transformation）は、複素数体 C 上の射影直線 P(C) に対する射影変換であるメビウス変換を指す用語として用いられる。より一般の数学的文脈において、複素数体 C はもっと別の環 (A, +, ×) に取り換えることができる。この場合の一次分数変換は、環 A 上の射影直線 P(A) 上の射影変換の意味である。A が可換環ならば、一次分数変換はよく知られた形 として書き表すことができるが、非可換の場合には右辺の点の座標をで (az + b, cz + d) と書くのが自然である。射影空間上の斉次座標の同値性に従えば、（cz + d が単元であるとき） が成り立つことに注意する。座標の同値性に従えば、（cz + d が単元であるとき） が成り立つことに注意する。 , Em matemática, uma transformação fracionár … Em matemática, uma transformação fracionária linear é, a grosso modo, uma transformação da forma que tem um inverso. As transformações fracionais lineares são amplamente utilizadas em várias áreas da matemática e suas aplicações na engenharia, como geometria clássica, teoria dos números (elas são usadas, por exemplo, na prova de Wiles do último teorema de Fermat), teoria dos grupos e teoria de controle. A definição precisa depende da natureza de a, b, c, d, e z.Em outras palavras, uma fracionária linear é uma transformação representada por uma fração cujo numerador e denominador são lineares. Na configuração mais básica, a, b, c, d, e z são números complexos (nesse caso, a transformação também é chamada de transformação de Möbius), ou mais geralmente elementos de um campo. A condição de inversibilidade é então ad – bc ≠ 0.Sobre um campo, uma transformação fracionária linear é a restrição ao campo de uma ou homografia da . Quando a, b, c, d são inteiros (ou, geralmente, pertencem a um domínio integral), z deve ser um número racional (ou pertencer ao corpo de frações do domínio integral. Nesse caso, a condição de inversibilidade é que ad – bc deve ser uma unidade do domínio (que é 1 ou -1 no caso de números inteiros). Na configuração mais geral, a, b, c, d e z são matrizes quadradas ou, mais geralmente, elementos de um anel. Um exemplo dessa transformação fracionária linear é a transformada de Cayley, que foi originalmente definida no anel matricial real 3 x 3.nte definida no anel matricial real 3 x 3. , Дробово-лінійне перетворення або дробово-лінійне відображення — це відображення комплексного простору на себе, яке здійснюється дробово-лінійними функціями. |
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| rdfs:comment | 数学の特に複素解析における一次分数変換（いちじぶんすうへんかん、英: linear … 数学の特に複素解析における一次分数変換（いちじぶんすうへんかん、英: linear fractional transformation）は、複素数体 C 上の射影直線 P(C) に対する射影変換であるメビウス変換を指す用語として用いられる。より一般の数学的文脈において、複素数体 C はもっと別の環 (A, +, ×) に取り換えることができる。この場合の一次分数変換は、環 A 上の射影直線 P(A) 上の射影変換の意味である。A が可換環ならば、一次分数変換はよく知られた形 として書き表すことができるが、非可換の場合には右辺の点の座標をで (az + b, cz + d) と書くのが自然である。射影空間上の斉次座標の同値性に従えば、（cz + d が単元であるとき） が成り立つことに注意する。座標の同値性に従えば、（cz + d が単元であるとき） が成り立つことに注意する。 , Дробово-лінійне перетворення або дробово-лінійне відображення — це відображення комплексного простору на себе, яке здійснюється дробово-лінійними функціями. , Em matemática, uma transformação fracionár … Em matemática, uma transformação fracionária linear é, a grosso modo, uma transformação da forma que tem um inverso. As transformações fracionais lineares são amplamente utilizadas em várias áreas da matemática e suas aplicações na engenharia, como geometria clássica, teoria dos números (elas são usadas, por exemplo, na prova de Wiles do último teorema de Fermat), teoria dos grupos e teoria de controle. A definição precisa depende da natureza de a, b, c, d, e z.Em outras palavras, uma fracionária linear é uma transformação representada por uma fração cujo numerador e denominador são lineares.cujo numerador e denominador são lineares. , Дро́бно-лине́йное преобразова́ние или дро́бно-лине́йное отображе́ние — это отображение комплексного пространства на себя, которое осуществляется дробно-линейными функциями. , In mathematics, a linear fractional transf … In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form which has an inverse. The precise definition depends on the nature of a, b, c, d, and z. In other words, a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator are linear. In the most general setting, the a, b, c, d and z are square matrices, or, more generally, elements of a ring. An example of such linear fractional transformation is the Cayley transform, which was originally defined on the 3 x 3 real matrix ring.lly defined on the 3 x 3 real matrix ring. , Lineární lomená funkce je funkce, kterou lze zapsat ve tvaru . |
| rdfs:label | Lineární lomená funkce , Linear fractional transformation , Transformação fracionária linear , Дробово-лінійне перетворення , Дробно-линейное преобразование , 一次分数変換 |
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