QR-DECOMPOSITION BY GRAM-SCHMIDT ORTHOGONALIZATION:
 random tall matrix A:
   0.9976   0.3674   0.3471
   0.5668   0.4806   0.3422
   0.9659   0.0738   0.2180
   0.7479   0.0054   0.1332
 check: Q*R: ==A?
   0.9976   0.3674   0.3471
   0.5668   0.4806   0.3422
   0.9659   0.0738   0.2180
   0.7479   0.0054   0.1332
 check: Q^T*Q: ==1?
   1.0000  -0.0000   0.0000
  -0.0000   1.0000   0.0000
   0.0000   0.0000   1.0000
 check: R: upper triangular?
   1.6759   0.4261   0.5074
   0.0000   0.4357   0.2125
   0.0000   0.0000   0.0157
 check: Q^T*A: ==R?
   1.6759   0.4261   0.5074
  -0.0000   0.4357   0.2125
   0.0000   0.0000   0.0157
 LINEAR SYSTEM OF EQUATIONS:
 random square matrix A:
   0.9005   0.6619   0.6464
   0.3868   0.0161   0.3230
   0.4455   0.6509   0.8557
 random right-hand-side v:
   0.4013   0.2069   0.9685
 solution x to A*x=v:
  -0.6171   0.1037   1.3743
 check: A*x: ==v?
   0.4013   0.2069   0.9685
 MATRIX INVERSE:
 inverse matrix A^-1:
   1.3771   1.0214  -1.4259
   1.3115  -3.3834   0.2864
  -1.7145   2.0417   1.6931
 check: A^-1*A: ==1?
   1.0000  -0.0000  -0.0000
  -0.0000   1.0000  -0.0000
   0.0000   0.0000   1.0000
 check: A*A^-1: ==1?
   1.0000   0.0000   0.0000
   0.0000   1.0000   0.0000
   0.0000   0.0000   1.0000