As in the 1 variable case the converse is not true in general : if all generalized Wronskians vanish, this does not imply that the functions are linearly dependent.
22.
In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation.
23.
In particular the Moore determinant vanishes if and only if the elements in the left hand column are linearly dependent over the finite field of order " q ".
24.
The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining a basis for a vector space.
25.
Therefore, we have d \ le wt ( \ boldsymbol { c'} ), which is the minimum number of linearly dependent columns in \ boldsymbol { H }.
26.
Each additional device has low odds of being linearly dependent with the existing set . ( roughly 1 / 2 ^ [ 40-dimensionality-of-spanned-space ] ).
27.
So two of these expressions must be the same which shows that log ? 1, . . ., log ? " n " are linearly dependent over the rationals.
28.
In linear algebra, a "'frame "'of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent.
29.
In over-/ under-determined linear systems, we may have many linearly dependent ( both mathematically and numerically ) rows which do not carry any additional information and hence are redundant.
30.
Since the second term in y _ 2 ( x ) is a scalar multiple of the first solution ( and thus linearly dependent ) we can drop that term, yielding a final solution of
