If the Af bond is linear then there are three reasonable positions for the hydrogen atoms: ( 1 ) The hydrogen atoms are centered and hence all lie on a sheet midway between the oxygen sheets ; ;

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If there is a bounded linear operator from X onto Y, then Y is reflexive.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

* If is the norm ( usually noted as ) defined in the sequence space ℓ < sup >∞</ sup > of all bounded sequences ( which also matches the non-linear distance measured as the maximum of distances measured on projections into the base subspaces, without requiring the space to be isotropic or even just linear, but only continuous, such norm being definable on all Banach spaces ), and is lower triangular non-singular ( i. e., ) then

If these are linear with constant

If the resulting linear differential equations have constant coefficients one can take their Laplace transform to obtain a transfer function.

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

If λ < sub > 1 </ sub >, ..., λ < sub > ν </ sub > are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others.

If I told you my son's age, then there would no longer be two unknowns ( variables ), and the problem becomes a linear equation with just one variable, that can be solved as described above.

Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

If the object is a vector space we have a linear representation.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

If the table size n is large enough, linear search will be faster than binary search, whose cost is O ( log n ).

If the list is stored as an ordered array, then binary search is almost always more efficient than linear search as with n > 8, say, unless there is some reason to suppose that most searches will be for the small elements near the start of the sorted list.

If, as she walked, her steps fumbled from time to time, she chose to ignore that omen.

If the bluff failed and they ran into trouble, Brannon had told the others, they would withdraw -- and he would come after his son another time.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If the would-be joiner asks these questions he is not likely to be duped by extremists who are seeking to capitalize on the confusions and the patriotic apprehensions of Americans in a troubled time.

If the world comes to this, wouldn't it be the very time when courage and American know-how would be needed to help survivors rebuild??

If their schedules were to synchronize, there was no point in wasting time.

`` If I catch you one more time down here without stockings '' --

If there is time after the warning, the basement shielding could be improved substantially by blocking windows with bricks, dirt, books, magazines, or other heavy material.

If, in the trustee's judgment, `` reasonable market conditions '' did not prevail during any given year, he was to be allowed to petition the court for an extension of time within the ten-year period.

If the reserve components are to serve effectively in time of war, their basic organization and objectives must conform to the changing character and missions of the active forces.

If you elect to use the Standard Deduction or the Tax Table, and later find you should have itemized your deductions, you may do so by filing an amended return within the time prescribed for filing a claim for refund.

If you mail A return or tax payment, you must place it in the mails in ample time to reach the district director on or before the due date.

If the lever is pulled to clear a signal or move a switch, the tappet moves a short distance lengthwise at the same time.

If necessary to replace both halves on grill, sear cuts and allot extra time.

If the background of design is too smooth, or you wish to create a wood-grained effect, it may be added at this time with a dull tool such as the handle of a fine paintbrush.

If companies will take the time to give objective consideration to their major problems and to the questions they provoke, then a long constructive step will have been taken toward more effective marketing in next decade.

If you've travelled in Europe a time or two, it is quite certain that you've had that wanting-to-be-alone feeling or that you will get it on your next visit across the Atlantic.

`` If you are trying to get us out of the brothel, the dustbin, the kitchen sink, and the tawdry living-room, you are probably wasting your time '', Moreland told me.

If, at any time during the assignment pass, the compiler finds that there are no more index words available for assignment, the warning message `` No More Index Words Available '' will be placed in the object program listing, the table will be altered to show that index words 1 through 96 are available, and the assignment will continue as before.

If Depew had told any academic psychologist that he had a weird feeling of having lived through that identical convention session at some time in the past, he would have been informed that he was a victim of deja vue.

If baby teeth are retained too long, the incoming second teeth may be prevented from emerging at the normal time or may have to erupt in the wrong place.

If one lives near a subway or an express parkway, the solution is to have one's wines stored with a dealer and brought home a few at a time.

If this woman had delayed until after 11:20 to start her shopping, she would have had little time in which to prepare the substantial meal that was eaten at dinner in those days.

If you have taken this stroll in the morning, and you have the time and inclination, walk to the right along the crowded Corso for half a dozen blocks to visit the fine private collection of paintings -- mainly of the sixteenth and seventeenth centuries -- in the Palazzo Doria ( open Sunday, Tuesday, and Thursday, 10:00 to 1:00 ).