If the direct-sum decomposition ( A ) is valid, how can we get hold of the projections Af associated with the decomposition??

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.

Statements such as the Banach – Tarski paradox can be rephrased as conditional statements, for example, " If AC holds, the decomposition in the Banach – Tarski paradox exists.

If sugars are heated so that all water of crystallisation is driven off, then caramelization starts, with the sugar undergoing thermal decomposition with the formation of carbon, and other breakdown products producing caramel.

If the fire was small, however, the convict would burn for some time until death from heatstroke, shock, the loss of blood and / or simply the thermal decomposition of vital body parts.

If I is a homogeneous ideal in A, then is also a graded ring, and has decomposition

If the LU decomposition is used, then the algorithm is unstable unless we use some sort of pivoting strategy.

If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an eigenvalue decomposition determined by solving the system of equations

If the LU decomposition exists, the LDU decomposition does too.

If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P = I, so an LU decomposition exists.

If the n eigenvalues are distinct ( that is, none is equal to any of the others ), then V is invertible, implying the decomposition.

If is the singular value decomposition of, then.

If the protocol is unconditionally concealing, then Alice can unitarily transform these states into each other using the properties of the Schmidt decomposition, effectively defeating the bindingness property.

If the matrix X < sup > T </ sup > X is well-conditioned and positive definite, that is, it has full rank, the normal equations can be solved directly by using the Cholesky decomposition R < sup > T </ sup > R, where R is an upper triangular matrix, giving:

If the measure ' is complex-valued i. e. is a complex measure, its upper and lower variation cannot be defined and the Hahn – Jordan decomposition theorem can only be applied to its real and imaginary parts.

If i ≠ j, then this decomposition does not contain the trivial representation ( Otherwise, we'd have a nonzero intertwiner from V < sub > j </ sub > to V < sub > i </ sub > contradicting Schur's lemma ).

If a split-complex number z does not lie on one of the diagonals, then z has a polar decomposition.

If they settle to the bottom, phytoplankton release ammonia during their decomposition, which returns nitrogen to the waters.

If the vacuum is degenerate and we have a mixed state, the cluster decomposition property fails.

If an integer n has prime factorization, then the primary decomposition of the ideal generated by, is

If cyanogen iodide is heated enough to undergo complete decomposition, it may releases toxic fumes of nitrogen oxides, cyanide and iodide.

If a matrix A is diagonalizable, the space can be decomposed into a direct sum of eigenspaces of A, and the matrix has an eigenspace decomposition.

If no specific organization plan exists limiting the number of scientists at each salary level, the result is a department top-heavy with high-level, high-salaried personnel ''.

If this be true, the possibility exists that an occlusive lesion of the bronchial arteries might cause widespread degeneration of supportive tissue similar to that seen in generalized emphysema.

If we cannot make explicit choices, how do we know that our set exists?

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

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 community exists, both freedom and security may exist as well.

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.

If the limit exists, then f is differentiable at a.

If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

If neither A nor B includes the idea of existence, then " some A are B " simply adjoins A to B. Conversely, if A or B do include the idea of existence in the way that " triangle " contains the idea " three angles equal to two right angles ", then " A exists " is automatically true, and we have an ontological proof of A's existence.

If total cash available is less than cash needs, a deficiency exists.

If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.

If it exists, the graviton is expected to be massless ( because the gravitational force appears to have unlimited range ) and must be a spin 2 boson.

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

If there exists an isomorphism between two groups, then the groups are called isomorphic.

If God exists in the understanding, we could imagine Him to be greater by existing in reality.

If such a function exists, we say X and Y are homeomorphic.

* 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 limit exists, we say that ƒ is complex-differentiable at the point z < sub > 0 </ sub >.

If an inverse function exists for a given function ƒ, it is unique: it must be the inverse relation.