Part I covers the basic concepts of linear algebra, including vectors and matrices,

linear independence and basis sets, determinants, eigenvalues, product spaces, and orthogonality, canonical forms, matrices with special properties such as Hermitian and unitary, and spectral theory.

The

linear independence of this new basis is easily proved using our theorem involving Hopf algebra calculus, while we have not been able to find an elementary proof of it.

A new set of

linear independence coefficient is generated to modify the coding vector when data packet reached the intermediate node.

Rivoal [7] showed a

linear independence result of polylogarithms, stated as follows.

introduced a scheme that utilizes a symmetric matrix construction for key distribution [8] that exploits the

linear independence merit of vectors to the solvability of linear systems (that is, difficulty of solving a system in n variables given t<n equations).

Canada) introduces the main ideas of functional analysis to beginning graduate or advanced undergraduate students with a prior knowledge of basic real analysis and such topics of elementary linear algebra as

linear independence, bases, and matrix manipulation.

An in-phase and quadrature time-invariant decomposition of Equation 2 is possible if it is assumed that the statistics of the resulting power spectral densities (PSD) are independent (in fact, the resulting PSDs are always independent due to

linear independence of sin and cos).

Using this notion of

linear independence, we define the notions of basis and dimension as in Guterman (2009); Cuninghame-Green and Butkovic (2004): a basis of a semimodule U over a semiring S is a set P of linearly independent elements from U which generate it, and the dimension of a semimodule U is the cardinality of its smallest basis.

Defect zero characters and relative defect zero characters Masafumi MURAI A property of the Fourier transform of probability measures on the real line related to the renewal theorem Yasuki ISOZAKI A note on

linear independence of polylogarithms over the rationals Noriko HIRATA-KOHNO and Hironori OKADA Above three, communicated by Masaki KASHIWARA, M.

The sixth edition introduces

linear independence and closure earlier in the text, and adds applications to air route analysis and the Google search engine.