Every proper rotation is the composition of two reflections, a special case of the Cartan – Dieudonné theorem.

Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R < sup > 3 </ sup > which is called the axis of rotation ( this is Euler's rotation theorem ).

Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number.

Every improper rotation of three-dimensional Euclidean space is rotation followed by a reflection in a plane through the origin.

Every rotation Rot ( φ ) has an inverse Rot (− φ ).

Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis ( Euler rotation theorem ).

Previously a daily segment, Underbelly didn't appear as often in the rotation ( hence Maddow's inclination to add " Every day ... or so " to her introduction of the segment ).

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.

* PASSIA Diaries and Annual Reports: Every year since 1988, PASSIA publishes its “ Diary ”, a unique annual resource book combining a comprehensive directory of contact information for Palestinian and interna tional institu tions operating in Palestine, a day-to-day calendar, and an agenda containing facts and figures, graphs, statistics, chronologies and maps related to Palestine and the Palestinians.

Every smooth manifold defined in this way has a natural diffeology, for which the plots correspond to the smooth maps from open subsets of R < sup > n </ sup > to the manifold.

Every covering map is a semicovering, but semicoverings satisfy the " 2 out of 3 " rule: given a composition of maps of spaces, if two of the maps are semicoverings, then so also is the third.

: Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.

Every measurable entity maps into a cardinal function but not every cardinal function is the result of the mapping of a measurable entity.

Every station has detailed maps of the station and surrounding area showing the locations of each exit.

Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram – Schmidt process.

Every building has a canonical length metric inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert space.

** Every vector space has a basis.

** Every field extension has a transcendence basis.

Every vector space has a basis, and all bases of a vector space have the same number of elements, called the dimension of the vector space.

Every finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ).

Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.

Every free abelian group has a rank defined as the cardinality of a basis.

For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ".

Many principles that imply the axiom of choice in their general form ( such as " Every vector space has a basis ") become provable in weak subsystems of second-order arithmetic when they are restricted.

Every vector a in three dimensions is a linear combination of the standard basis vectors i, j, and k.

Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients.

Every software development methodology approach acts as a basis for applying specific frameworks to develop and maintain software.

Every faculty, school and department has its own administrative body, the members of which are democratically elected on the basis of collective processes.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

Every simple R-module is isomorphic to a quotient R / m where m is a maximal right ideal of R. By the above paragraph, any quotient R / m is a simple module.

Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.

Every Boolean ring R satisfies x ⊕ x

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

* Every left ideal I in R is finitely generated, i. e. there exist elements a < sub > 1 </ sub >, ..., a < sub > n </ sub > in I such that I = Ra < sub > 1 </ sub > + ... + Ra < sub > n </ sub >.

* Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion.

Every year since 1982, the W. C. Handy Music Festival is held in the Florence / Sheffield / Muscle Shoals area, featuring blues, jazz, country, gospel, rock music and R & B.

Every adult citizen of this small settlement signed the small petition ; E. K Dyer and his wife, William Johnson, Joseph Otis and his wife, Hiram Walker and his wife, Joseph Pease and R. H. Valentine.

Every smooth submanifold of R < sup > n </ sup > has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on R < sup > n </ sup >.