By Franklin A. Graybill

Regression research: options and functions makes a speciality of pondering basically approximately and fixing functional statistical difficulties. The procedure leads from the theoretical (meaning conceptual no longer mathematical) to the utilized, the belief being that samples (using conception) inform the investigator what should be identified approximately populations (for application).

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Second, third 1—38. Determine the rank of (a) 1 3 7 5 2 4 —4 —10 3 (b) 2 3 —1 4 6 —2 —2 —3 1 1 2 —1 1—39. Let a be of order in x ii and rank r. Show that a has n — r columns which are linear combinations of r linear independent columns. Verify for 1234 a= 2 5 1 3 2 7 12 14 1—40. Using properties 3, 4, and 7 of determinants (see Sec. 1—7), deduce that the elementary operations do not change the rank of a matrix. For con- venience, consider the first r rows and columns to be linearly independent.

The defect for this system is 1. If a is of rank 1, the second row is a scalar multiple, say A, of the first row. Multiplying the second equation in (1—64) by 1/A, we have a12x2 + a13x3 = a11x1 + a12x2 + a13x3 = a11x1 + C1 c2/A (1—69) 2cr, the equations are inconsistent and no solution exists. Then, when a is of rank 1, (1—64) has a solution only if the rows of c are related in the same manner as the rows of a. If this condition is satisfied, the two equations in (1—69) are identical and one can be disregarded.

1—20, 1—21. f See Ref. 4, sect. 3 15, for a discussion of the general Laplace expansion method. The expansion in terms of cofactors for a iow Or a COlUmn is a special case of the general method. CRAMER'S RULE SEC. 1—9. The evaluation of a determinant, using the definition equation (1—44) or the cofactor expansion formula (1—46) is quite tedious, particularly when the array is large. A number of alternate and more efficient numerical procedures for evaluating determinants have been developed. These procedures are described in References 9—13.