Help


[permalink] [id link]
+
Page "Bra-ket notation" ¶ 100
from Wikipedia
Edit
Promote Demote Fragment Fix

Some Related Sentences

For and finite-dimensional
For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:
For this reason, it was proposed that the original finite-dimensional space be mapped into a much higher-dimensional space, presumably making the separation easier in that space.
For any finite-dimensional G-invariant subspace V in L < sup > 2 </ sup >( G ), where G acts on the left, we consider the image of G in GL ( V ).
For the most reasonable finite-dimensional spaces ( such as compact manifolds, finite simplicial complexes or CW complexes ), the sequence of Betti numbers is 0 from some points onwards ( Betti numbers vanish above the dimension of a space ), and they are all finite.
Note that if R is a finite-dimensional simple algebra over a division ring E, D need not be contained in E. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.
For finite-dimensional vector spaces, this agrees with the previous definition
For a system in which the space H is finite-dimensional, entropy is maximized for the states S which in diagonal form have the representation
For the finite-dimensional complex Hilbert space, one writes
For every finite-dimensional vector space ( viewed as a module over the base field ), the length and the dimension coincide.
For a general bilinear form B on a finite-dimensional vector space over any field we can define a Gram matrix G attached to a set of vectors by.
For a finite-dimensional vector space V, the dual space V * is isomorphic to V, and for any given set of basis vectors
For an irreducible finite-dimensional representation, the highest weight can't be less than the lowest weight.
For the case of a finite-dimensional graph ( having a finite number of edges and vertices ), the discrete Laplace operator is more commonly called the Laplacian matrix.
( For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.
* For a field k, a finite-dimensional, unital, associative algebra is Frobenius if and only if the injective right A-module Hom < sub > k </ sub >( A, k ) is isomorphic to the right regular representation of A.
For a positive selfadjoint operator S on a finite-dimensional Euclidean space V, the formula

For and vector
For vector spaces over non-algebraically closed fields, we still need to find some substitute for characteristic values and vectors.
For each picture frame, the 6502 writes graphics commands for the DVG into a defined area of RAM ( the vector RAM ), and then asks the DVG to draw the corresponding vector image on the screen.
For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.
For a vector with linear addressing, the element with index i is located at the address B + c · i, where B is a fixed base address and c a fixed constant, sometimes called the address increment or stride.
For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:
For any system, the following restrictions on measurement results apply, where is the reduced Planck constant and is any direction vector such as x, y, or z:
For 2D vector animations, the rendering process is the key frame illustration process, while tweened frames are rendered as needed.
For example, binary string 1011 is represented by the vector ( 1, 0, 1, 1 ).
For the purposes of this article, we call this constructed vector the transmitted vector.
For a vector expressed in cylindrical coordinates as
For twisting rotation to exist, it is necessary to tilt the body sideways after takeoff, so that there is now a small component of this horizontal angular momentum vector along the body's long axis.
Malaria Foundation International states, " The outcome of the treaty is arguably better than the status quo going into the negotiations ... For the first time, there is now an insecticide which is restricted to vector control only, meaning that the selection of resistant mosquitoes will be slower than before.
For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:
For a flow, the vector field Φ ( x ) is a linear function of the position in the phase space, that is,
For a more useful solution, we utilize vector identities, which work for any vector, as follows:
For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.
For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.
For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

For and space
For example, even the most successful executive lives in a two-room apartment while ordinary people rent space in the stairwells of office buildings in which to sleep at night ; ;
The Industrial Division published, in 1960, a new, attractive industrial brochure, `` Rhode Island -- Right For Industry '', and prepared copy for a new edition of the Directory Of Manufacturers ( to be printed shortly ), and for a new space catalogue.
For example, consider the sample space
* James ` s Theorem For a Banach space the following two properties are equivalent:
For example, in three-dimensional complex Euclidean space,
For ƒ ∈ C ( X ) ( with a compact Hausdorff space X ), one sees that:
For example, a shopping mall does not provide the merchandise a shopper is seeking, but provides space and services for retailers that serve the shopper.
For instance, each photon that travels through intergalactic space loses energy due to the redshift effect.
For the study of the mechanical behavior of solids and fluids these are assumed to be continuous bodies, which means that the matter fills the entire region of space it occupies, despite the fact that matter is made of atoms, has voids, and is discrete.
For instance, any continuous function defined on a compact space into an ordered set ( with the order topology ) such as the real line is bounded.
For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà – Ascoli theorem held in the sense of mean convergence — or convergence in what would later be dubbed a Hilbert space.
For instance, a diagonal operator on the Hilbert space may have any compact nonempty subset of C as spectrum.
For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:
For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.
For example, in AC power lines, the waves of electromagnetic energy propagate through the space between the wires, moving from a source to a distant load, even though the electrons in the wires only move back and forth over a tiny distance.
For impacts into highly porous materials, a significant crater volume may also be formed by the permanent compaction of the pore space.
For any topological space X the ( Alexandroff ) one-point compactification αX of X is obtained by adding one extra point ∞ ( often called a point at infinity ) and defining the open sets of the new space to be the open sets of X together with the sets of the form G < font face =" Arial, Helvetica "> U </ font >
For example, when an ordinary RGB digital image is compressed via the JPEG standard, the RGB colorspace is first converted ( by a rotation matrix ) to a YCbCr colorspace, because the three components in that space have less correlation redundancy and because the chrominance components can then be subsampled by a factor of 2 or 4 to further compress the image.
For example, the conservation of energy follows from the time-invariance of physical systems, and the fact that physical systems behave the same regardless of how they are oriented in space gives rise to the conservation of angular momentum.
For example, a user space program cannot delete a file ; it can delete a link to a file, and if the kernel determines that there are no hard links to the file, it may then allow the memory location for the deleted file to be allocated for another file.
For example, NASA's workhorse space shuttle used cryogenic hydrogen / oxygen propellant as its primary means of getting into orbit, and all of the rockets built for the Soviet space program by Sergei Korolev used liquid oxygen as their oxidiser.

0.162 seconds.