where b and c are scalars, v is any vector of size n and I is the identity matrix of size n. These properties state that the determinant is an alternating multilinear function of the columns, and they suffice to uniquely calculate the determinant of any square matrix.

These algorithms need only a few multiplications and additions to calculate each vector.

These are extended objects that are charged sources for differential form generalizations of the vector potential electromagnetic field.

These derivations form a real vector space if we define addition and scalar multiplication for derivations by

These numbers are the coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system, and are typically called the scalar components ( or scalar projections ) of the vector on the axes of the coordinate system.

These numbers are often arranged into a column vector or row vector, particularly when dealing with matrices, as follows:

These probability amplitudes can be represented as a complex number or equivalent vector — or, as Richard Feynman simply calls them in his book on QED, " arrows ".

These random segments are inserted into a plasmid or bacteriophage vector, which is in turn implanted into Escherichia coli bacteria.

These two pieces of information can be represented as a 2-dimensional vector, as a complex number, or as magnitude ( amplitude ) and phase in polar coordinates.

These include theorems about compactness of certain spaces such as the Banach – Alaoglu theorem on the compactness of the unit ball of the dual space of a normed vector space, and the Arzelà – Ascoli theorem characterizing the sequences of functions in which every subsequence has a uniformly convergent subsequence.

These implemented an instruction set similar to the Fujitsu VP2000 vector supercomputer and had a nominal performance of 200 megaflops on double precision arithmetic and double that on single precision.

These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials.

The so-called classical groups are subgroups of GL ( V ) which preserve some sort of bilinear form on a vector space V. These include the

( These feature vectors can be seen as defining points in an appropriate multidimensional space, and methods for manipulating vectors in vector spaces can be correspondingly applied to them, such as computing the dot product or the angle between two vectors.

These are often written using normal vector notation ( e. g. i, or ) rather than the circumflex notation, and in most contexts it can be assumed that i, j, and k, ( or and ) are versors of a Cartesian coordinate system ( hence a term of mutually orthogonal unit vectors ).

These residuals are further treated to reduce but not eliminate pathogens and vector attraction by any of a number of approved methods and then trucked and land appied to a farm field.

These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or modules.

Any covector field α has components in the basis of vector fields f. These are determined by

These ambiguities are resolved by the right-hand rule: With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area dS.

These two requirements show that stochastic vectors have a geometric interpretation: A stochastic vector is a point on the " far face " of a standard orthogonal simplex.

These materials then become anisotropic, and Young's modulus will change depending on the direction of the force vector.

These act on smooth vector fields to yield smooth functions by pointwise evaluation, namely, given a covector field ω and a vector field X, we define

These fields frequently overlap, but tend to use different methodologies and techniques.

These are made from special ceramics in which mechanical vibrations and electrical fields are interlinked through a property of the material itself.

These unmodified fields are to provide a location to harbor pests.

These cylindrical carbon molecules have unusual properties, which are valuable for nanotechnology, electronics, optics and other fields of materials science and technology.

These fields have produced prominent figures within these two industries.

These forces arise from the presence of the body in force fields, e. g. gravitational field ( gravitational forces ) or electromagnetic field ( electromagnetic forces ), or from inertial forces when bodies are in motion.

These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.

These fields of study were essentially created by Claude Shannon, who published fundamental papers on the topic in the late 1940s and early 1950s.

These are sometimes called ( especially in technical fields ) the major and minor semi-axes, the major and minor semiaxes, or major radius and minor radius.

These E and B fields are also in phase, with both reaching maxima and minima at the same points in space ( see illustrations ).

These derivatives require that the E and B fields in EMR are in-phase ( see math section below ).

These fields and spacing between fields increase from the dorso-lateral MEA to the ventro-medial MEA.

* Linear algebraic groups ( or more generally affine group schemes ) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different ( and much less well understood ).

These fields of view covered most angles from 0 to 180 degrees, fanning out from the spin axis.

These fields are active areas of research in inorganic chemistry, aimed toward new catalysts, superconductors, and therapies.

These simple relationships hold true only when there is no mutual coupling of magnetic fields between individual inductors.

These fields are interrelated often and share the common goals of protecting the confidentiality, integrity and availability of information ; however, there are some subtle differences between them.

These protected for a time the authors in all scientific fields such as printers, copper engravers, map makers and publishers.

These differences in relief, together with stretches of water interspersed with forests, fields, and pastures are the main features that make the landscape so distinctive.

These fields comprise both pure mathematics and applied mathematics and establish links between the two.

These standards specify the familiar formats for text email headers and body and rules pertaining to commonly used header fields such as " To :", " Subject :", " From :", and " Date :".

These fields, generated by passing electric currents through gradient coils, make the magnetic field strength vary depending on the position within the magnet.

These fields in turn underlie modern electrical and communications technologies.