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Polynomial Interpolation
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
The columns of the result of the null command give a basis for the null space of the matrix. But the solution set of Ax=0 is precisely null(A), so any element of the null set is a solution of the equation. In particular, any basis vector is a nontrivial solution. (If A is square and nonsingular, null(A) contains only the zero vector and hence has no basis.) Hence to solve Eq. (5.8), say, we can find null(C), which must have at least one basis vector (that is, dimnull(C)≥1) and we may use that basis vector as our nontrivial solution q to Cq=0.
Elements of Quantum Electronics
Published in Michael Olorunfunmi Kolawole, Electronics, 2020
Before we delve into qubits in state space, it is important to define some notations used in quantum mechanics as compared to classical system. The space of states in classical system is Boolean. Analogous to its Boolean counterpart, the space of states in quantum system is not a mechanical set: it is a vector space (also called Hilbert space). A vector space, say V, over the field of complex numbers ℂ is a non-empty set of elements called vectors. As such, in V, it is defined as the operations of vector addition and multiplication of a vector by a scalar in ℂ. A vector space may have either a finite or an infinite number of dimensions, but in most applications in quantum computation, finite vector spaces are used for completeness; denoted by ℂn. The dimension of a vector space is the number of basis vectors.
Vector Spaces
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
In this section we will show that for a given m × n matrix AThe number of vectors in a basis for the column space of A is equal to the number of vectors in a basis for the row space of A. This number is called the rank of A.The number of vectors in a basis for the null space of A is called the nullity of A. We will show that nullity of A + rank of A = number of columns of A.
An improved version of conditioned time and frequency domain reverse path methods for nonlinear parameter estimation of MDOF systems
Published in Mechanics Based Design of Structures and Machines, 2023
In contrast to the frequency domain approach, the conditioning in the time domain approach is performed directly in the time domain. Therefore, the errors involved in the transformation of time domain data into a frequency domain with windowing can be completely avoided. The conditioning process in time domain approach is also recursive and results in where the conditioned response is obtained by projecting the actual response onto the subspace of the Euclidean space which is orthogonally spanned by the nonlinear terms and indicate the coefficient vector and are a set of orthonormal vectors spanned by the nonlinear terms (Muhamad, Sims, and Worden 2012). The orthonormal basis is computed using the Gram-Schmidt orthogonalization process (Hoffmann 1989). Equation (26) can be rewritten in recursive form as