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.

* ( Soundness ) Every provable second-order sentence is universally valid, i. e., true in all domains under standard semantics.

* ( Completeness ) Every universally valid second-order formula, under standard semantics, is provable.

Every second-order linear ODE with three regular singular points can be transformed into this equation.

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 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 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 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 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 vector in the space may be written as a linear combination of unit vectors.

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 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 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.

Every day, when the President took his nap, Rob Roy would stretch out on the window seat near him, like a perfect gentleman, and stare thoughtfully out the window, or he would take a little nap himself.

Every man who dabbles in the market to make a little easy money on the side and suffers losses could at the time hardly face his wife who was wondering how her husband could be so dumb.

Every eye was on him as he began to speak.

Every few minutes she would awaken for a moment to review things: Stowey, yes, was on his way south, and the two boys were away in school, and nothing was burning on the stove, and Lucretia was coming for dinner and bringing three guests of hers.

Every night they all went to Mama and Papa Albright's, and sat on the open front porch, where they could get the breeze.

Every single problem touched on thus far is related to good marketing planning.

Every few days, in the early morning, as the work progressed, twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement, supported as it is by ancient brick vaulting.

Every decision was made quickly on sound grounds.

Every night when he wanted a drink of water, didn't he practice being fearless by not turning on the bathroom light??

Every single instance of altruistic behavior need not always increase inclusive fitness ; altruistic behaviors would have been selected for if such behaviors on average increased inclusive fitness in the ancestral environment.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

Every year, on the last Sunday in April, there is an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth.

Every actor who has opened in the role of Pseudolus on Broadway ( Zero Mostel, Phil Silvers and Nathan Lane ) won a Best Actor Tony Award for their performance.

* Revised Code of Washington 9. 12. 010: " Every person who brings on his or her own behalf, or instigates, incites, or encourages another to bring, any false suit at law or in equity in any court of this state, with intent thereby to distress or harass a defendant in the suit, or who serves or sends any paper or document purporting to be or resembling a judicial process, that is not in fact a judicial process, is guilty of a misdemeanor ; and in case the person offending is an attorney, he or she may, in addition thereto be disbarred from practicing law within this state.

Every cladogram is based on a particular dataset that was analyzed with a particular method.

Every year, the European Joint Conferences on Theory and Practice of Software ( ETAPS ) sponsors the International Conference on Compiler Construction ( CC ), with papers from both the academic and industrial sectors.

Every Halloween, Dabney House conducts the infamous " Millikan pumpkin-drop experiment " from the top of Millikan Library, the highest point on campus.

# Every net on X has a convergent subnet ( see the article on nets for a proof ).

# Every filter on X has a convergent refinement.

# Every ultrafilter on X converges to at least one point.

* Pseudocompact: Every real-valued continuous function on the space is bounded.

Every person should be identified uniquely as resident in one place but where they happen to be on census day, their de facto residence, may not be the best place to count them.