Fri Apr  4 09:17:51 2025

rref2_test():
  python version: 3.10.12
  numpy version:  1.26.4
  Test rref(), which analyzes matrices using the
  reduced row echelon form (RREF)

is_rref_test():
  is_rref() reports whether a matrix is in reduced row echelon format.

  A0:
[[1 0 0 9 4]
 [0 0 1 0 8]
 [0 0 0 0 0]
 [0 0 0 1 0]]
  is_rref(A0) =  False

  A1:
[[1 0 0 9 4]
 [0 0 0 1 0]
 [0 0 1 0 8]
 [0 0 0 0 0]]
  is_rref(A1) =  False

  A2:
[[1 0 0 9 4]
 [0 1 0 2 8]
 [0 0 3 0 0]
 [0 0 0 0 0]]
  is_rref(A2) =  False

  A3:
[[1 0 3 9 4]
 [0 1 0 2 8]
 [0 0 1 0 0]
 [0 0 0 0 0]]
  is_rref(A3) =  False

  A4:
[[1 0 3 0 4]
 [0 1 2 0 8]
 [0 0 0 1 0]
 [0 0 0 0 0]]
  is_rref(A4) =  True

rref_columns_test():
  rref_columns() uses the reduced row echelon form (RREF)
  of a matrix to find the linearly independent columns.

  Matrix A:
[[ 1  2  3  1]
 [ 2  4  9  3]
 [ 3  6  0  0]
 [ 4  8  0  2]
 [ 5 10  6  6]
 [ 6 12  6  3]
 [ 7 14  2  1]]

  Number of independent columns is  3

  Independent columns of A:
[[1 3 1]
 [2 9 3]
 [3 0 0]
 [4 0 2]
 [5 6 6]
 [6 6 3]
 [7 2 1]]

  Matrix columns: