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Some Related Sentences
Every and vector

**
Every vector space has
a basis
.
Every subset A
of the
vector space is contained within
a smallest convex set ( called the convex hull
of A ), namely the intersection
of all convex sets containing A
.
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 normed
vector space V sits
as a dense subspace inside
a Banach space ; this Banach space is essentially uniquely defined by V and is called the completion
of V
.
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 random
vector gives rise
to a probability measure on
R < sup > n </ sup > with the Borel
algebra as the underlying sigma-algebra
.
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 vector in the space may
be written
as a linear combination
of unit vectors
.
Every finite-dimensional Hausdorff topological
vector space is reflexive
, because J is bijective by
linear algebra, and because there is
a unique Hausdorff
vector space topology on
a finite dimensional
vector space
.
Every coalgebra
, by (
vector space ) duality
, gives rise
to an
algebra, but not
in general the other way
.

*
Every holomorphic
vector bundle on
a projective variety is induced by
a unique algebraic
vector bundle
.

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 space is free
, and the free
vector space on
a set is
a special case
of a free module on
a set
.
Every and v
Every one
of the infinitely many vertices
of G
can be reached
from v < sub >
1 </ sub > with
a simple path
, and each such path must start with one
of the finitely many vertices adjacent
to v < sub >
1 </ sub >.
Every maximal outerplanar graph satisfies
a stronger condition than Hamiltonicity: it is node pancyclic
, meaning that for every vertex
v and every k
in the range
from three
to the number
of vertices
in the graph
, there is
a length-k cycle containing
v. A cycle
of this length may
be found by repeatedly removing
a triangle that is connected
to the rest
of the graph by
a single edge
, such that the removed vertex is not
v, until the outer face
of the remaining graph has length k
.
Every Vitali set V is uncountable
, and
v − u is irrational for any
.

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Every vector space V with seminorm p (
v ) induces
a normed space V / W
, called the quotient space
, where W is the subspace
of V consisting
of all vectors
v in V with p (
v )

When
a variable
, v, is on the LHS
of an assignment statement
, such
as s ( j ), then s ( j ) is
a definition
of v. Every variable (
v ) has at least one definition by its declaration ( V ) ( or initialization ).
Every and determines
Every telephone company
, whether large or small
, determines its own ANAC for each individual central office
, which tends
to perpetuate the current situation
of a mess
of overlapping and / or spotty areas
of coverage
.
Every action has
a reaction and the force
determines one's next incarnation
.
Every number is
thought of as a decimal fraction with the initial decimal point omitted
, which determines the filing order
.

:
Every host
in the network must have
a unique address that
determines where it is
.
Every structure is associated with
a certain quantity
of energy
, which determines the stability
of the molecule or ion ( the lower energy
, the greater stability ).
Every telephone company
, whether large or small
, determines its own ringback numbers for each individual central office
, which tends
to perpetuate the current situation
of a mess
of overlapping and / or spotty areas
of coverage
.
Every determines one probability distribution for
.
Every two years
, the MSHSL
determines a school's activity classification and section placement
.
Every character has three stats
which help him
in battle: Brawn stat
determines weapon damage
, Brains affects the effectiveness
of super powers
, and Toughness is the character's defense
.

*
Every pair
of distinct points
determines a unique line
.

*
Every non-collinear set
of n points
determines at least n distinct lines
.
Every March
, a 68-team
, six-round
, single-elimination tournament ( commonly called March Madness )
determines the national champions
of NCAA Division I men's college basketball
.
Every and linear
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 time
a diode switches
from on
to off or vice versa
, the configuration
of the
linear network changes
.
Every smooth ( or differentiable )
map φ: M → N between smooth ( or differentiable ) manifolds induces natural
linear maps between the corresponding tangent spaces:

*
Every linear combination
of its components Y =
a < sub >
1 </ sub > X < sub >
1 </ sub > + … +
a < sub > k </ sub > X < sub > k </ sub > is normally distributed
.
Every sedenion is
a real
linear combination
of the unit sedenions
1, < var > e </ var >< sub >
1 </ sub >, < var > e </ var >< sub > 2 </ sub >, < var > e </ var >< sub > 3 </ sub >, ..., and < var > e </ var >< sub > 15 </ sub >,
Every octonion is
a real
linear combination
of the unit octonions:
Every physical quantity has
a Hermitian
linear operator associated
to it
, and the states where the value
of this physical quantity is definite are the eigenstates
of this
linear operator
.
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 finite-dimensional normed space is reflexive
, simply because
in this case
, the space
, its dual and bidual all have the same
linear dimension
, hence the
linear injection J
from the definition is bijective
, by the rank-nullity theorem
.
Every real m-by-n matrix yields
a linear map from R < sup > n </ sup >
to R < sup > m </ sup >.

*
Every ( biregular ) algebraic automorphism
of a projective space is projective
linear.
Every bounded
linear transformation
from a normed
vector space
to a complete
, normed
vector space
can be uniquely extended
to a bounded
linear transformation
from the completion
of to.
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 linear program has
a dual problem with the same optimal solution
, but the variables
in the dual problem correspond
to constraints
in the primal problem and vice versa
.

*
Every irreducible polynomial
in K
which has
a root
in L factors into
linear factors
in L
.
Every linear function on
a finite-dimensional space is continuous
.
0.977 seconds.