ambient(Matrix) -- get the map between the ambient free modules

i1 : R = QQ[a..d];

i2 : f = map(image vars R, coker matrix{{a,b},{c,d}}, transpose matrix{{a,b,c,d},{d,c,b,a}})

o2 = {1} | a 2d |
     {1} | b 2c |
     {1} | c 0  |
     {1} | d 0  |

o2 : Matrix

i3 : target f

o3 = image | a b c d |

                             1
o3 : R-module, submodule of R

i4 : source f

o4 = cokernel | a b |
              | c d |

                            2
o4 : R-module, quotient of R

i5 : ambient f

o5 = | a2+b2+c2+d2 2bc+2ad |

             1      2
o5 : Matrix R  <-- R