However, in practice ( as the calculations are performed in floating point arithmetic where inaccuracy is inevitable ), the orthogonality is quickly lost and in some cases the new vector could even be linearly dependent on the set that is already constructed.
42.
At very short time scales, however, the motion of a particle is dominated by its inertia and its displacement will be linearly dependent on time : ? " x " = " v " ? " t ".
43.
In analysing the dynamics of wheelsets and complete rail vehicles, the contact forces can be treated as linearly dependent on the creep ( Kalker's linear theory, valid for small creepage ) or more advanced theories can be used from frictional contact mechanics.
44.
Determinants have other applications, including a systematic way of seeing if a set of vectors is linearly independent ( we write the vectors as the columns of a matrix, and if the determinant of that matrix is zero, the vectors are linearly dependent ).
45.
Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane.
46.
Has rank 2 : the first two rows are linearly independent, so the rank is at least 2, but all three rows are linearly dependent ( the third is equal to the second subtracted from the first ) so the rank must be less than 3.
47.
In practice, one of them acts as a main ( independent ) voltage source with voltage " V 1 " and the other as an additional ( linearly dependent ) voltage source with voltage V _ 2 = K { V _ 1 }.
48.
The frame condition was first described by Richard Duffin and Albert Charles Schaeffer in a 1952 article on nonharmonic Fourier series as a way of computing the coefficients in a linear combination of the vectors of a linearly dependent spanning set ( in their terminology, a " Hilbert space frame " ).
49.
To test whether the third equation is linearly dependent on the first two, postulate two parameters " a " and " b " such that " a " times the first equation plus " b " times the second equation equals the third equation.
50.
The vectors b, Ab, \ ldots A ^ { n-1 } b might be almost linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors q _ 1, q _ 2, \ ldots, q _ n \, which form a basis for K _ n.
