By J J Connor
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This publication contains lecture notes of a summer time institution named after the overdue Jacques Louis Lions. The summer time university used to be designed to alert either Academia and to the expanding function of multidisciplinary tools and instruments for the layout of complicated items in quite a few parts of socio-economic curiosity.
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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.