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tangent basis e 1, e 2, e 3 to the coordinate curves (left), dual basis, covector basis, or reciprocal basis e 1, e 2, e 3 to coordinate surfaces (right), in 3-d general curvilinear coordinates (q 1, q 2, q 3), a tuple of numbers to define a point in a position space. Note the basis and cobasis coincide only when the basis is orthonormal. [1 ...
is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinates of the new basis vectors on the old basis. A change of basis is sometimes called a change of coordinates, although it excludes many coordinate transformations.
For example, the standard basis on R n provides a standard orientation on R n (in turn, the orientation of the standard basis depends on the orientation of the Cartesian coordinate system on which it is built). Any choice of a linear isomorphism between V and R n will then provide an orientation on V. The ordering of elements in a basis is crucial.
When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but the basis has, so the components of the vector must change to compensate.
Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884. In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
However, Rv produces a rotation in the opposite direction with respect to wR. Throughout this article, rotations produced on column vectors are described by means of a pre-multiplication. To obtain exactly the same rotation (i.e. the same final coordinates of point P), the equivalent row vector must be post-multiplied by the transpose of R (i.e ...
Basis (or cost basis), as used in United States tax law, is the original cost of property, adjusted for factors such as depreciation. When a property is sold, the taxpayer pays/(saves) taxes on a capital gain /(loss) that equals the amount realized on the sale minus the sold property's basis.
Heraclitus also uses the succession of opposites as a basis for change: Cold things grow hot, hot things grow cold, a moist thing withers, a parched thing is wetted. (DK B126) An object persists despite opposite properties, even as it undergoes change.