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Outermorphism
In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function. Definition Let f be an \mathbb-linear map from V to W. The extension of f to an outermorphism is the unique map \textstyle \underline : \bigwedge(V) \to \bigwedge(W) satisfying : \underline(1) = 1 : \underline(x) = f(x) : \underline(A \wedge B) = \underline(A) \wedge \underline(B) : \underline(A + B) = \underline(A) + \underline(B) for all vectors x and all multivectors A and B, where \textstyle \bigwedge(V) denotes the exterior algebra over V. That is, an outermorphism is a unital algebra homomorphism between exterior algebras. The outermorphism inherits linearity properties of the original linear map. For example, we see that for scalars \alpha, \beta and vectors x, y, z, the outermorphism is li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometric Algebra
In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not by all elements) and addition of objects of different dimensions. The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Gras ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Zero Map
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently division by zero has no meaning in arithmetic. As a numerical digit, 0 plays a crucial role in decimal notation: it indicates that the power of ten corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies in place-value notations that uses a base other than ten, such as binary and hexadecimal. The modern use of 0 in this manner derives from Indian mathematics that was transmitted to Europe via medieval Islamic mathematicians and popularized by Fibonacci. It was independently used by the Maya. Common names for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rotor (mathematics)
A rotor is an object in the geometric algebra (also called Clifford algebra) of a vector space that represents a rotation about the origin. The term originated with William Kingdon Clifford, in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre). Hestenes Hestenes uses the notation R^\dagger for the reverse. defined a rotor to be any element R of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies R\tilde R = 1, where \tilde R is the "reverse" of R—that is, the product of the same vectors, but in reverse order. Definition In mathematics, a rotor in the geometric algebra of a vector space ''V'' is the same thing as an element of the spin group Spin(''V''). We define this group below. Let ''V'' be a vector space equipped with a positive definite quadratic form ''q'', and let Cl(''V'') be the geometric algebra associated to ''V''. The algebra Cl(''V'') i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Eigenvalues And Eigenvectors
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Up To
Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, " is unique up to " means that all objects under consideration are in the same equivalence class with respect to the relation . Moreover, the equivalence relation is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation that relates two lists if one can be obtained by reordering (permutation, permuting) the other. As another example, the statement "the solution to an indefinite integral is , up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. A pseudoscalar, when multiplied by an ordinary vector, becomes a '' pseudovector'' (or ''axial vector''); a similar construction creates the pseudotensor. A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a pseudovector. In physics In physics, a pseudoscalar denotes a physical quantity analogous to a scalar. Both are physical quantities which assume a single value which is invariant under proper rotations. However, under the parity transformation, pseudoscalars flip their signs while scalars do not. As reflections through a plane ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Underline
An underscore or underline is a line drawn under a segment of text. In proofreading, underscoring is a convention that says "set this text in italic type", traditionally used on manuscript or typescript as an instruction to the printer. Its use to add emphasis in modern finished documents is generally avoided. The (freestanding) underscore character, , also called a low line, or low dash, originally appeared on the typewriter so that underscores could be typed. To produce an underscored word, the word was typed, the typewriter carriage was moved back to the beginning of the word, and the word was overtyped with the underscore character. In modern usage, underscoring is achieved with a markup language, with the Unicode combining low line or as a standard facility of word processing software. The free-standing underscore character is used to indicate word boundaries in situations where spaces are not allowed, such as in computer filenames, email addresses, and in Internet U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Transpose
In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. Transpose of a matrix Definition The transpose of a matrix , denoted by , , , A^, , , or , may be constructed by any one of the following methods: #Reflection (mathematics), Reflect over its main diagonal (which runs from top-left to bottom-right) to obtain #Write the rows of as the columns of #Write the columns of as the rows of Formally, the -th row, -th column element of is the -th row, -th column element of : :\left[\mathbf^\operatorname\right]_ = \left[\mathbf\right]_. If is an matrix, then is an matrix. In the case of square matrices, may also denote the th power of the matrix . For avoiding a possibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
