Let x and y be

linearly independent unit vectors in our Hubert space and P and Q the projections onto the subspaces they generate.

We provide an algorithm in terms of linear algebra to determine whether or not H is nondegenerate over F: For any hyperplane H [subset] [P.sup.n](F) with coefficients in F; Let L be the linear form in n+1 variables defining H, that is [Mathematical Expressions Omitted], where P is the projection map; For , a set of linear forms in n+1 variables which are pairwise

linearly independent, we denote by [(L).sub.F] the vector space generated by the linear forms in L over F.

Both (2) and (3) comprise N x [N.sub.j]

linearly independent, non-homogeneous equations for the [N.sub.j] firms in N industries and have, therefore, unique solutions.

Once the ciphertexts of 2n

linearly independent message vectors are known, the encryption function is uniquely defined.

These packets, which are defined over [F.sup.n.sub.q], arrive at the sink nodes where they must be

linearly independent so that the system has a solution.

Suppose that the set L([[alpha].sub.1], ..., [[alpha].sub.n]) is

linearly independent over the field of rational numbers Q, and [c.sub.1], ..., [c.sub.n] are complex numbers, at least one of them is non-zero.

Also three

linearly independent solutions of the field equations in (1), namely, of the system of equations (9), are given by

The new training data subset is generated by the

linearly independent base transformation.

Firstly, we will show that S is a

linearly independent subset of H [union] {0}.

Notice that the geometric multiplicity is the largest number of

linearly independent eigenvectors associated with an eigenvalue.

Second calculate the all-possible paths of the TFG based on formula which is the possible paths equal the number of

linearly independent paths.

The hyperplanes [H.sub.1], ..., [H.sub.k] in A are

linearly independent if their normals are

linearly independent.