We carried out the linear independence
analysis of thermal parameters, the impact of the U-pipe shank spacing value on the effective borehole thermal resistance and calculated errors of thermal parameters under various TRT durations.
(c) To establish the linear independence
between the codes of flows (packets) arriving at a sink node t [member of] T, a matrix [P.sub.t] is constructed whose rows contain these codes.
For example, if [[chi].sub.1] and [[chi].sub.2] are non-equivalent characters, and the numbers [[alpha].sub.1] and [[alpha].sub.2] are algebraically independent over Q, then the function L(s, [[chi].sub.1]) L(s, [[chi].sub.2]) [zeta](s, [[alpha].sub.1]) [zeta](s, [[alpha].sub.2]) satisfies the hypotheses of Theorem 8, since a polynomial p(s) has a preimage (1,1,1,p(s)) [member of] [S.sup.2] x [H.sup.2](D), and the algebraic independence of the numbers [[alpha].sub.1] and [[alpha].sub.2] implies the linear independence
of the set L(P, [[alpha].sub.1], [[alpha].sub.2]).
One of the key differences between the deterministic linear network coding technique (DLNC) and the RLNC is that the RLNC coding coefficients increase the probability of their linear independence
. It is also shown in  that the larger the Galois Field, the higher the probability of linear independence
of its elements (coding coefficients) as concisely summarised in Table 1.
To find out the connection between the performances in the two tests we applied the test of linear independence
based on the table above.
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.
This method ranks the contribution of each candidate sensor location to the linear independence
of the corresponding target modes; it also maximizes the determinant of Fisher information matrix.
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.
Further improvements include the formalization of linear combinations, linear dependence, linear independence
, and subspace.
Rivoal  showed a linear independence
result of polylogarithms, stated as follows.