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: Any linear transformation f: V → A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T ( V ) to A as indicated by the following commutative diagram:
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Any and linear
Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.
:“ Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T ( V ) to A .”
Any linear polarization of light can be written as an equal combination of right-hand ( RHC ) and left-hand circularly ( LHC ) polarized light:
Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:
# Any sequence satisfying the recurrence relation can be written uniquely as a linear combination of solutions constructed in part 1 as λ varies over all distinct roots of p ( t ).
* Any linear functional L is either trivial ( equal to 0 everywhere ) or surjective onto the scalar field.
Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence ( hence " the " canonical divisor ).
A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ ( y )= 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g /.
Any smooth function of one variable approximates a quadratic function when examined near enough to its minimum point ; and therefore the force — which is the derivative of energy with respect to displacement — will approximate a linear function.
Any anisotropic or isotropic linear algebraic group over a field becomes split over the algebraic closure, so this distinction is interesting from the point of view of Algebraic number theory.
Any linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line.
Any point that is rigidly connected to the body can be used as reference point ( origin of coordinate system L ) to describe the linear motion of the body ( the linear position, velocity and acceleration vectors depend on the choice ).
* Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.
Any linear code can be represented as a graph, where there are two sets of nodes-a set representing the transmitted bits and another set representing the constraints that the transmitted bits have to satisfy.
Any linear time-invariant operation on s ( t ) produces a new spectrum of the form H ( f )• S ( f ), which changes the relative magnitudes and / or angles ( phase ) of the non-zero values of S ( f ).
Any system in a large class known as linear, time-invariant ( LTI ) is completely characterized by its impulse response.
Any other sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U < sub > n </ sub >( P, Q ) and V < sub > n </ sub >( P, Q ).
Any linear system containing only voltage sources, current sources, and other resistors can be converted to a Thévenin equivalent circuit, comprising exactly one voltage source and one resistor representing " internal resistance ".
Any Jacobi field can be represented in a unique way as a sum, where is a linear combination of trivial Jacobi fields and is orthogonal to, for all.
Any and transformation
Any conformal map on a portion of Euclidean space of dimension greater than 2 can be composed from three types of transformation: a homothetic transformation, an isometry, and a special conformal transformation.
Any transformation of the plane leaving this pattern invariant can be decomposed into a translation, ( x, y ) → ( n + x, y ), optionally followed by a reflection in either the horizontal axis, ( x, y ) → ( x ,− y ), or the vertical axis, ( x, y ) → (− x, y ), provided that this axis is chosen through or midway between two dots, or a rotation by 180 °, ( x, y ) → (− x ,− y ) ( ditto ).
Any waveform can be disassembled into its spectral components by Fourier analysis or Fourier transformation.
Any segments of the enterprise that are not service-oriented can also be documented in order to consider transformation requirements if a service needs to communicate with the business processes automated by such segments.
Any canonical transformation involving a type-2 generating function G < sub > 2 </ sub >( q, P, t ) leads to the relations
Any two triplets obeying these relationships must be related by a linear transformation ; they represent utility indices differing only by scale and origin.
Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space.
Any and f
Note: Any nonlinear electronic block driven by two signals with frequencies f < sub > 1 </ sub > and f < sub > 2 </ sub > would generate intermodulation ( mixing ) products.
Any frequency component above f < sub > s </ sub >/ 2 is indistinguishable from a lower-frequency component, called an alias, associated with one of the copies.
Any group can be seen as a category with a single object in which every morphism is invertible ( for every morphism f there is a morphism g that is both left and right inverse to f under composition ) by viewing the group as acting on itself by left multiplication.
* quantification Any morphism f: X → Y in a category with pullbacks induces a monotonous map acting by pullbacks ( A monotonous map is a functor if we consider the preorders as categories ).
Any function passed as an argument to f is invoked with itself as an argument, and thus in any use of that argument is indirectly referring to itself.
A graph coloring is an assignment of one of k colors to a graph G so that the endpoints of each edge have different colors, for some number k. Any coloring corresponds to a homomorphism from G to a complete graph K < sub > k </ sub >: the vertices of K < sub > k </ sub > correspond to the colors of G, and f maps each vertex of G with color c to the vertex of K < sub > k </ sub > that corresponds to c. This is a valid homomorphism because the endpoints of each edge of G are mapped to distinct vertices of K < sub > k </ sub >, and every two distinct vertices of K < sub > k </ sub > are connected by an edge, so every edge in G is mapped to an adjacent pair of vertices in K < sub > k </ sub >.
Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, a nucleus, or an electron, when placed in an external magnetic field B ( measured in teslas ) that is not aligned with its magnetic moment, will precess at a frequency f ( measured in hertz ), that is proportional to the external field:
Any section s of E over B induces a section of f < sup >*</ sup > E, called the pullback section f < sup >*</ sup > s, simply by defining.
Any non-vanishing holomorphic function f defined on the strip can be approximated by the ζ-function.
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