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...monics a power of ''r''² cannot be factored out by forming some linear combination [since regular harmonics span an [[irreducible representation]] of the [[gr A contracted set is a predetermined linear combination of radial parts of GTOs (that are on the same center). Indexing different p

...''a''₁, ..., ''a''_''p'' not all zero, such that the linear combination is the zero vector in ''V'',

...accounted for, and we can take it out of this space. The single remaining linear combination will belong to another eigenspace of '''J'''². We can repeat this argument for the three products with ''M'' = 3. One linear combination of the three is already accounted for and can be removed, two linear combin

*see also [[Linear combination of atomic orbitals molecular orbital method]]

...independent]], it follows that any vector in the plane can be written as a linear combination of these two non-collinear vectors. (This is also expressed as: Any vector

...independent]], it follows that any vector in the plane can be written as a linear combination of these two non-collinear vectors. (This is also expressed as: Any vector

...e of the system. In other words, the state of any system is reducible to a linear combination of solutions of the Schrödinger equation for that system. The Hamiltonian ...y exist in the system as defined, and so the particle's state must be some linear combination of the eigenstates of ''H''.

...the same longitudinal vector. Hence, the three fields can be written as a linear combination of two orthonormal vectors, '''e'''_''x'' and '''e'''<sub>''y''</

It can be shown that the following linear combination is the only totally invariant function with ''ℓ'' = 4:

...mponents of the same second rank tensor, a Lorentz transformation gives a linear combination of '''E''' and '''B''', and hence in relativity theory these two fields do

Any 2x2 Hermitian matrix may be written as a linear combination of the 2&times;2 identity matrix and the three [[Pauli spin matrices]]. The

...m variables called '''factors'''. The observed variables are modeled as [[linear combination]]s of the factors, plus "[[errors and residuals in statistics|error]]" term ...r constant times their level of mathematical intelligence, i.e., it is a [[linear combination]] of those two "factors". The numbers, for this particular subject, by wh

into the linear combination

{{Image|Moving wave.gif|right||<small>Fig. 5. Linear combination of lowest 3 modes </small> }}

...that the electron wavefunction in a crystal is made up of a [[LCAO method|linear combination of atomic orbitals]] with undetermined coefficients. Symmetry of the crysta

...component of '''A''' is a vector perpendicular to '''k''' and hence is a linear combination of '''e'''⁽¹⁾ and '''e'''^(&minus;1). The superscript

...''A''-th nucleus, then the MO &phi; of electron 1 has the following LCAO (linear combination of atomic orbitals) form,

...pread out" over a molecule. Usually this is a consequence of an MO being a linear combination (weighted sum) of atomic orbitals centered on different atoms.

<math>\mathbf{r}_3</math>. Thus the latter two vectors are a linear combination of the first two,