Linear Algebra and the C Language/a08h

__NOTOC__ Install and compile this file in your working directory. /* ------------------------------------ */ /* Save as : c00b.c */ /* ------------------------------------ */ #include "v_a.h" /* ------------------------------------ */ #define RA R4 #define CA C6 #define Cb C1 /* ------------------------------------ */ #define CbFREE Cb+C5 /* ------------------------------------ */ int main(void) { double ab[RA*(CA+Cb)]={ +9, -15, +21, -18, +6, +27, +0, -18, +30, -42, +36, -12, -54, +0, +21, -35, +49, -42, +14, +63, +0, -6, +10, -14, +12, -4, -18, +0 }; double **Ab = ca_A_mR(ab, i_Abr_Ac_bc_mR(RA,CA,Cb)); double **A = c_Ab_A_mR(Ab, i_mR(RA,CA)); double **b = c_Ab_b_mR(Ab, i_mR(RA,Cb)); double **Ab_free = i_Abr_Ac_bc_mR(CA,CA,CbFREE); double **b_free = i_mR(CA,CbFREE); double **A_bfree = i_mR(RA,CbFREE); int r; clrscrn(); printf("Find a basis for the orthogonal complement of A :\n\n"); printf(" A :"); p_mR(A,S6,P1,C10); printf(" b :"); p_mR(b,S6,P1,C10); printf(" Ab :"); p_mR(Ab,S6,P1,C10); stop(); clrscrn(); printf(" Ab : gj_PP_mR(Ab,NO) :"); gj_PP_mR(Ab,NO); p_mR(Ab,S7,P3,C10); put_zeroR_mR(Ab,Ab_free); printf(" Ab_free : put_zeroR_mR(Ab,Ab_free);"); p_mR(Ab_free,S7,P3,C10); put_freeV_mR(Ab_free); printf(" Ab_free : put_freeV_mR(Ab_free);"); p_mR(Ab_free,S7,P3,C10); stop(); clrscrn(); r = rsize_R(Ab_free); while(r>R1) zero_above_pivot_gj1Ab_mR(Ab_free,r--); printf(" Ab_free : zero_above_pivot_gj1Ab_mR(Ab_free,r--);"); p_mR(Ab_free,S7,P3,C10); c_Ab_b_mR(Ab_free,b_free); printf(" b_free is a basis for the orthogonal complement of A"); p_mR(b_free,S10,P3,C7); stop(); clrscrn(); printf(" A :"); p_mR(A,S10,P3,C10); printf(" b_free is a basis for the orthogonal complement of A"); p_mR(b_free,S10,P3,C7); printf(" The row vectors of A" " are orthogonal to the column vectors of bfree\n\n"); printf(" A * bfree: "); p_mR(mul_mR(A,b_free,A_bfree),S10,P3,C7); stop(); f_mR(Ab); f_mR(A); f_mR(b); f_mR(Ab_free); f_mR(b_free); f_mR(A_bfree); return 0; } /* ------------------------------------ */ /* ------------------------------------ */ We begin by calculating the free variables. The columns of b_free form a basis for the orthogonal complement of A. A * b_free = 0 This proves that the row vectors of A are orthogonal to the column vectors of b_free. Screen output example: Find a basis for the orthogonal complement of A : A : +9.0 -15.0 +21.0 -18.0 +6.0 +27.0 -18.0 +30.0 -42.0 +36.0 -12.0 -54.0 +21.0 -35.0 +49.0 -42.0 +14.0 +63.0 -6.0 +10.0 -14.0 +12.0 -4.0 -18.0 b : +0.0 +0.0 +0.0 +0.0 Ab : +9.0 -15.0 +21.0 -18.0 +6.0 +27.0 +0.0 -18.0 +30.0 -42.0 +36.0 -12.0 -54.0 +0.0 +21.0 -35.0 +49.0 -42.0 +14.0 +63.0 +0.0 -6.0 +10.0 -14.0 +12.0 -4.0 -18.0 +0.0 Press return to continue. Ab : gj_PP_mR(Ab,NO) : +1.000 -1.667 +2.333 -2.000 +0.667 +3.000 +0.000 +0.000 +0.000 -0.000 +0.000 +0.000 +0.000 +0.000 +0.000 -0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 -0.000 +0.000 +0.000 +0.000 +0.000 Ab_free : put_zeroR_mR(Ab,Ab_free); +1.000 -1.667 +2.333 -2.000 +0.667 +3.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 Ab_free : put_freeV_mR(Ab_free); +1.000 -1.667 +2.333 -2.000 +0.667 +3.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +1.000 Press return to continue. Ab_free : zero_above_pivot_gj1Ab_mR(Ab_free,r--); +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.667 -2.333 +2.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 -0.667 -3.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +1.000 b_free is a basis for the orthogonal complement of A +0.000 +1.667 -2.333 +2.000 -0.667 -3.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 Press return to continue. A : +9.000 -15.000 +21.000 -18.000 +6.000 +27.000 -18.000 +30.000 -42.000 +36.000 -12.000 -54.000 +21.000 -35.000 +49.000 -42.000 +14.000 +63.000 -6.000 +10.000 -14.000 +12.000 -4.000 -18.000 b_free is a basis for the orthogonal complement of A +0.000 +1.667 -2.333 +2.000 -0.667 -3.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +1.000 The row vectors of A are orthogonal to the column vectors of bfree A * bfree: +0.000 -0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 -0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 -0.000 +0.000 +0.000 +0.000 Press return to continue.