It follows that the set-composition K can be recovered from the monomial of lexicographically largest evaluation in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies the linear independence of the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have not been able to prove the linear independence of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] without using our theorem to duplicate the alphabet.

The proof of Theorem A is based on an important linear independence criterion due to Yu.

We rely on Theorem C and Theorem D instead of Nesterenko's linear independence criterion, by following Rivoal's argument of Pade approximation [7].

Zudilin, A refinement of Nesterenko's linear independence criterion with applications to zeta values, Math.

Hata, On the linear independence of the values of polylogarithmic functions, J.

Given the aforementioned

linear independence property, Eve can ensure that [Y.

Linear independence over tropical semirings and beyond.

He introduces vectors and matrices in data mining and pattern recognition, then gives more details on vectors and matrices, including their inner products, vector norms and

linear independence bases.

Appendices cover complex numbers, basic matrix operations, determinants and a review of the basics in spanning,

linear independence, basis and dimension, and change of basis.