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 α on tangent vectors can be expanded in terms of the dual basis θ

* 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 ).

* Any linear function is both concave and convex.

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 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:

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 linear function is homogeneous of degree 1, since by the definition of linearity

Any linear combination

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 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 approach toward such a system looks toward saving and security.

Any type of focal lesion of the central nervous system ( such as stroke, brain tumour, multiple sclerosis ) will cause the type of ataxia corresponding to the site of the lesion: cerebellar if in the cerebellum, sensory if in the dorsal spinal cord ( and rarely in the thalamus or parietal lobe ), vestibular if in the vestibular system ( including the vestibular areas of the cerebral cortex ).

Any heat that isn't translated into work is normally considered a waste product and is removed from the engine either by an air or liquid cooling system.

Any customer who opted to keep theirs was required to sign a waiver indicating their understanding that no more software would be written for the system and which absolved Intellivision of any future responsibility for technical support.

: Premise: Any cognitive state occurs as part of a set of cognitive states and within a cognitive system

Any individual workflow within a system of workflows can be subject to some type of operational latency.

Any object gravitationally held by the rotating Earth – Moon system will be attracted to the barycenter to an equal and opposite degree as its tendency to fly off into space.

Any communication to another system in the galaxy has to be recorded on a physical disc which is transported through the wormhole ; chains of these are referenced in Diplomatic Immunity ( Chapter 9 ) where messages are ' squirted ' to a station on one side of a wormhole, flown ( on a physical medium ) through the wormhole and re-squirted to a similar arrangement at the next wormhole in the sequence.

Eugene Odum, a founder of ecology, stated: " Any unit that includes all of the organisms ( ie: the " community ") in a given area interacting with the physical environment so that a flow of energy leads to clearly defined trophic structure, biotic diversity, and material cycles ( i. e.: exchange of materials between living and nonliving parts ) within the system is an ecosystem.

Any real mode operating system ( such as Xenix ) could also be made to run using OS / 2's virtual machine capabilities, subject to certain direct hardware access limitations.

* Extensibility: Any database system has its own predefined set of data types which we can use to build new types ( see Complex Objects ), but a purely extensible system is designed such that there are no logical differences between built-in ( or ' primitive ') data types and ' custom ' data types, though there may be strong differences in the way the two are implemented.

Any decidable type system involves a trade-off: while it rejects many incorrect programs, it can also prohibit some correct, albeit unusual programs.

Any two-level quantum system can be used as a qubit.

* Any system that implements records management

Any automated reasoning system will have to deal with truly huge inputs.

: Any change in status quo prompts an opposing reaction in the responding system.

Any decrease in the Gibbs function of a system is the upper limit for any isothermal, isobaric work that can be captured in the surroundings, or it may simply be dissipated, appearing as T times a corresponding increase in the entropy of the system and / or its surrounding.

Any CTCSS system that has compatible tones is interchangeable.

Any object radiating sound, including a loudspeaker system, can be thought of as being composed of combinations of such simple point sources.

Any direction can potentially serve as the plane of reference, as long as it is clearly defined for everyone using that system.

Any security system depends crucially on keeping some things secret.

Any foreigner wishing to visit Ming China could only do so via the tribute system.