Orthogonality ( hence also orthonormality ) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees linear independence of the functions.
12.
If the eigenvalues have an algebraic multiplicity greater than 1, then repeat the process, but now multiplying by an extra factor of for each repetition, to ensure linear independence.
13.
When the \ ell'replace the \ ell, both the linear independence and spanning properties for infinite basis sets follow from present " outer condition " and conversely.
14.
His work on matroids culminated in the paper " Representation of matroids " published in 1969 . In his work Ingleton studied matroids as a generalization of the concept of linear independence.
15.
This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of matroids, called linear matroids and algebraic matroids respectively.
16.
An important application is to compute linear independence : a set of vectors is linearly independent if and only if the Gram determinant ( the determinant of the Gram matrix ) is non-zero.
17.
I can get to working out what the formulas would be, and I think you just assume linear independence between the error terms and the independent variable, allowing the substitutions to be easier.
18.
On the contrary, linear independence is a mere instance of present independence, which becomes equivalent to it in such vector spaces . ( Also, several other generalizations of linear independence for universal algebras do not imply present independence .)
19.
On the contrary, linear independence is a mere instance of present independence, which becomes equivalent to it in such vector spaces . ( Also, several other generalizations of linear independence for universal algebras do not imply present independence .)
20.
One new point is computed on each iteration, usually by solving a trust region subproblem subject to the bound constraints, or alternatively, by choosing a point to replace an interpolation point so as to promote good linear independence in the interpolation conditions.
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