Since any vector x may be written as , it follows that scalars are unaffected with .

Subsequently, the distance order of the feature vectors to the given data vector x is determined.

Notice that the vector x is an eigenvector to the corresponding eigenvalue λ.

The vector x is real and non-null but otherwise arbitrary.

Thus a vector x satisfying (33) has s arbitrary elements.

To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector x 1.

The numbers a are called the coordinates of the vector x with respect to the basis B, and by the first property they are uniquely determined.

In statistics, a random vector x is classically represented by a probability density function.

Two similar notations are used for the Euclidean norm of a vector x:

Here, the notation x 0 means that all components of the vector x are nonnegative.