A Tale of Two Fractals by A.A. Kirillov

Author:A.A. Kirillov
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(4.2.6)

It is well known that the determinant of the Gram matrix of a system of n vectors in equals the square of the determinant consisting of the coordinates of these vectors. The same is true up to sign for pseudo-Euclidean spaces, e.g., for .

Since , we have . It follows that the vectors p i are linearly independent, and hence they form a basis in .

For any vector , we define its covariant coordinates v i and contravariant coordinates v j with respect to the basis {p i } as follows:

(4.2.7)

Let us find the relationship between these coordinates. From Eqs. (4.2.6) and (4.2.7), we have

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