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Geometry
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
If V is a three-dimensional vector space over the field of real numbers, then, thinking of the points of P(V) in terms of homogenous coordinates relative to a given fixed ordered basis of V, the points [0:1:0] and [0:0:1] lie on a unique line, which is the span of these vectors (i.e., the two-dimensional subspace of V consisting of all points with homogenous coordinates with first component zero). In the dual space of V, relative to the dual basis of the given ordered basis of V, this subspace corresponds to the subspace of all linear functionals of V that annihilate all such vectors. This is clearly the span of the first vector in the ordered dual basis. Thus, the homogenous line coordinates for the line containing the two given points is [1:0:0]. This can also be obtained as a cross product.
An Introductory Review of Quantum Mechanics
Published in Ramaswamy Jagannathan, Sameen Ahmed Khan, Quantum Mechanics of Charged Particle Beam Optics, 2019
Ramaswamy Jagannathan, Sameen Ahmed Khan
As in the finite-dimensional case, it is possible to construct a dual basis for a nonorthonormal basis. If a set of functions {ξj(x)} form a nonorthonormal basis, then a set of dual basis functions {ξ˜j(x)} can be defined such that
Scalars and Vectors
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Thus, not only is the tangent vector basis orthogonal, it is also parallel to the dual basis, with the only difference between the two being a factor of ha2. This also means that we could just as well construct the orthonormal basis starting from the dual basis.
A mixed variational formulation for a class of contact problems in viscoelasticity
Published in Applicable Analysis, 2018
A. Matei, S. Sitzmann, K. Willner, B. I. Wohlmuth
In order to get the discrete version of the problem, we define to be the number of all nodes in the discrete domain. The nodes are divided into three sets, the potential contact nodes on the slave side as a discretization of , the potential contact nodes on the second body, also called master nodes , and all other nodes . We denote by the standard nodal basis function for node in and by the dual basis function for all , . The nodal dual basis functions are linear combinations of the nodal standard basis functions and are constructed with the biorthogonality condition Defining , the displacements are now discretized with standard basis functions and the Lagrange multipliers with dual basis functions . The indices n for time discretization and i for current time increment are omitted here for better readability.
Tensor calculus: unlearning vector calculus
Published in International Journal of Mathematical Education in Science and Technology, 2018
Wha-Suck Lee, Johann Engelbrecht, Rita Moller
Most functional analysts would not call the Jacobian a dualism map. The term duality has a specific meaning: given a basis (e1, e2) of , where , the dual basis (e*1, e*2) consists of linear functionals on V where for each fixed j, for j = 1, 2. Each fixed e*j ∈ V* can be represented by a fixed vector vj in V in the sense that for j = 1, 2. So, the representation (5.2) allows us to identify V* with V. For the purpose of distinguishing the basis from the dual basis, we use subscript and superscript notations: the dual basis of the basis (e*1, e*2) is denoted by (e1, e2) and the map is called the dualism map. Functional analysts would not be up in arms to call E a dualism map. The Gram matrix formed by dotting the basis vectors plays a critical role in computing the representation vectors vi of Equation (5.2). We can identify its inverse Eij with the dualism map E: The Einstein notation once again contracts (5.4) in a more suggestive form: The constructs (5.1)–(5.5) carry over to the derivative basis (Z1, Z2) of tensor calculus (Section 4). Tensor calculus calls the derivative basis (Z1, Z2) the covariant basis and the dual basis (Z1, Z2) the contravariant basis. The matrix inverse Zij of the Gram matrix for the covariant basis plays the role of a dualism map Z: Zi↦Zi: We shall use the notation Z and Zij interchangeably. Note that the dualism map Zij is local to coordinate system of choice: