| #### D.3.1.23 colred Procedure from library `matrix.lib` (see matrix_lib). Usage: colred(A[,e]); A matrix, e any type Return: - a matrix B, being the column reduced form of A, if colred is called with one argument. (as far as this is possible over the polynomial ring; no division by polynomials) - a list L of two matrices, such that L[1] = A * L[2] with L[1] the column-reduced form of A and L[2] the transformation matrix (if colred is called with two arguments). Assume: The entries of A are in the base field. It is not verified whether this assumption holds. Note: * This procedure is designed for teaching purposes mainly. * It applies rowred to the transposed matrix. proc gauss_col should be faster. * It should only be used with exact coefficient field (there is no pivoting) over the polynomial ring (ordering lp or dp). * Parameters are allowed. Hence, if the entries of A are parameters the computation takes place over the field of rational functions. Example:
    LIB "matrix.lib";
    ring r=(0,a,b),(A,B,C),dp;
    matrix m[8][6]=
    0,    2*C, 0,    0,  0,   0,
    0,    -4*C,a*A,  0,  0,   0,
    b*B,  -A,  0,    0,  0,   0,
    -A,   B,   0,    0,  0,   0,
    -4*C, 0,   B,    2,  0,   0,
    2*A,  B,   0,    0,  0,   0,
    0,    3*B, 0,    0,  2b,  0,
    0,    AB,  0,    2*A,A,   2a;"";
    ==>
    print(colred(m));"";
    ==> 0,0,0,0,    2*C, 0,
    ==> 0,0,0,0,    -4*C,(a)*A,
    ==> 0,0,0,(b)*B,-A,  0,
    ==> 0,0,0,-A,   B,   0,
    ==> 1,0,0,0,    0,   0,
    ==> 0,0,0,2*A,  B,   0,
    ==> 0,1,0,0,    0,   0,
    ==> 0,0,1,0,    0,   0
    ==>
    list L=colred(m,1);
    print(L[1]);
    ==> 0,0,0,0,    2*C, 0,
    ==> 0,0,0,0,    -4*C,(a)*A,
    ==> 0,0,0,(b)*B,-A,  0,
    ==> 0,0,0,-A,   B,   0,
    ==> 1,0,0,0,    0,   0,
    ==> 0,0,0,2*A,  B,   0,
    ==> 0,1,0,0,    0,   0,
    ==> 0,0,1,0,    0,   0
    print(L[2]);
    ==> 0,        0,         0,     1,        0,               0,
    ==> 0,        0,         0,     0,        1,               0,
    ==> 0,        0,         0,     0,        0,               1,
    ==> 1/2,      0,         0,     2*C,      0,               -1/2*B,
    ==> 0,        1/(2b),    0,     0,        -3/(2b)*B,       0,
    ==> -1/(2a)*A,-1/(4ab)*A,1/(2a),-2/(a)*AC,(-2b+3)/(4ab)*AB,1/(2a)*AB
See also: gauss_col.

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