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If V and W are finite-dimensional, then the space of all linear transformations from W to V, denoted Hom ( W, V ), is generated by such outer products ; in fact, the rank of a matrix is the minimal number of such outer products needed to express it as a sum ( this is the tensor rank of a matrix ).
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If and V
If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.
If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.
If D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.
If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.
If and we have for all v, w in V, then we say that B is symmetric.
If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.
If is a ket in V and is a ket in W, the direct product of the two kets is a ket in.
If an epidemic of cholera is suspected, the most common causative agent is V. cholerae O1.
If V. cholerae serogroup O1 is not isolated, the laboratory should test for V. cholerae O139.
If V is finite-dimensional, then V * has the same dimension as V. Given a basis
If the electrode has a positive potential with respect to the SHE, then that means it is a strongly reducing electrode which forces the SHE to be the anode ( an example is Cu in aqueous CuSO < sub > 4 </ sub > with a standard electrode potential of 0. 337 V ).
* If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application defined by is continuous.
* 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 V is a set of strings then V * is defined as the smallest superset of V that contains λ ( the empty string ) and is closed under the string concatenation operation.
# If V is a set of symbols or characters then V * is the set of all strings over symbols in V, including the empty string.
If and W
If it has a width of W pixels and a height of H pixels we say that the frame size is WxH.
The evidence was compiled by W de Sitter ( 1927 ) who wrote " If we accept this hypothesis, then the ' astronomical time ', given by the earth's rotation, and used in all practical astronomical computations, differs from the ' uniform ' or ' Newtonian ' time, which is defined as the independent variable of the equations of celestial mechanics ".
If a ship's hull classification symbol begins with " W ", it is a ship of the United States Coast Guard.
If μ is an unknown constant and W is random noise with probability density, then has probability density and is therefore a location family ..
If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories ( T, U, V, W, Z ) count fourfold, I counts twice, and X counts only once.
If there are p of these parts of the rope supporting the load W, then a force balance on the moving block shows that the tension in each of the parts of the rope must be W / p.
If W is irreducible then each type can, through a sequence of mutations ( i. e. powers of W ) mutate into all the other types.
If W isn't irreducible, then the dominant species ( or quasispecies ) that develops can depend on the initial population, as is the case in the simple example given below.
If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.
If n = 2k is even, then let W ′ be an isotropic subspace complementary to W. If n = 2k + 1 is odd let W ′ be a maximal isotropic subspace with W ∩ W ′
*" If I Could Tell You " by W. H.
If and are
If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;
If I even hint at it do you think it will matter that you are his nephew -- and not even a blood nephew ''??
If the circumstances are faced frankly it is not reasonable to expect this to be true.
If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.
Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.
If Jews are identified as a religious body in a controversy that comes before a national or international tribunal, it is obviously compatible with the goal of human dignity to protect freedom of worship.
If they are right, they will prevail of and by themselves.
Without preliminaries, Esther asked him, `` If you are a world citizen, will you take Garry Davis' place in his tent while he goes to the hospital ''??
If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.
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 we break the minister to our bit, we are buying back our own sins.
If the record buyer's tastes are somewhat eclectic or even the slightest bit esoteric, he will find them satisfied on educational records.
If Daddy's books are out of bounds his own picture books are not.
If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.
If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??
If the Communists are sincere in wanting a united, neutral and disarmed Germany, it might well be advantageous for the German people in this nuclear age.
If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.
If any are left, presently, we may expect to see signs specifically prohibiting the feeding of them too.
If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.
If it is not enough that all of our internationalist One Worlders are advocating that we join this market, I refer you to an article in the New York Times' magazine section ( Nov. 12, 1961 ), by Mr. Eric Johnston, entitled `` We Must Join The Common Market ''.
If the UN troops are not mercenaries then the Hessians were not mercenaries either.
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