Cannot get conditional Take[...] from List[...] to work inside a user-defined function

Question

Abstract I have a function eqsfromMatrix[...] that does exactly as it sounds. It returns a list of equations given matrix m for different augmented vectors in b so that eqs = m.vars == b. It takes last few letters in alphabet to form the variables i.e. {x, y, z}. If b not supplied but set to b = 1 in arguments it takes the first few letters in alphabet to form scalers b = {a,b,c} or if b set in arguments as b = 0 it will supply list of zeroes b = {0, 0, 0,...}. It starts using letters for variables where I set a pivot which is a desired soft start point "x".

Problem is it won't work if the length of the matrix m is longer than the position of the pivot in the supplied alphabet.

About the Code If using m1 and b1 and I replace vars = Take[abc, {start[m], end[m]}]; with vars = Take[abc, {23, 26}]; it works just fine.

What am I doing wrong?

eqsfromMatrix[mat_, b_ : None, piv_ : "x"] := 
 Module[{len, abc, pivot, diff, vars, scals, eqs},
  len = Length[mat];
  abc = Alphabet[];
  pivot = Flatten[Position[abc, piv]][[1]]; (* desired soft start point at x returns 24 *)
  start[m_] := pivot /; len <= 3;
  start[m_] := Length[abc] - len + 1;
  end[m] = start[m] + len - 1;
  vars = Part[abc, {start[m], end[m]}];
  eqs = mat.vars;
  scals = 
   Switch[b, None, Return[eqs], _List, b, 1, Take[abc, {1, len}], 0, 
    Table[0, len]];
  Table[eqs[[i]] == scals[[i]], {i, 1, len}]
  ]

ClearAll[m1,m2,b1,b2]
m1 = {{1, 0, 1, 3}, {-1, 3, 2, 1}, {3, 2, 4, 5}, {8, 3, 5, 3}};
b1 = {-1, 3, 2, 4};(* m1, b1 work , length is longer than pivot position from end of alphabet *)
m2={{1,2},{3,2}} 
b2={1,-2}(* m2, b2 work, length is shorter than pivot position from end of 
eqsfromMatrix[m1] (* dont work *)
eqsfromMatrix[m1,b1] (* dont work *)
eqsfromMatrix[m1,1] (* dont work *)
eqsfromMatrix[m1,0] (* dont work *)
eqsfromMatrix[m2] (* works *)
eqsfromMatrix[m2,b2] (* works *)
eqsfromMatrix[m2,1] (* works *)
eqsfromMatrix[m2,0] (* works *)

Answer 1

Your code block with the fixes:

ClearAll[eqsfromMatrix]
eqsfromMatrix[mat_, b_ : None, piv_ : "x"] := Module[{len = Length[mat],
    abc = Alphabet[], pivot, vars, scals, eqs, start, end},
  pivot = Flatten[Position[abc, piv]][[1]];
  start = If[len <= 3, pivot, Length[abc] - len + 1];
  end = start + len - 1;
  vars = Part[abc, start ;; end]; (* or Take[abc, {atart, end}]; *)
  eqs = mat.vars;
  scals = Switch[b, None, Return[eqs], _List, b, 
    1, Take[abc, {1, len}], 0, Table[0, len]];
  Table[eqs[[i]] == scals[[i]], {i, 1, len}]]

eqsfromMatrix[m1]

 {"w" + "y" + 3 "z", -"w" + 3 "x" + 2 "y" + "z", 
  3 "w" + 2 "x" + 4 "y" + 5 "z", 8 "w" + 3 "x" + 5 "y" + 3 "z"}

eqsfromMatrix[m1, b1] 

 {"w" + "y" + 3 "z" == -1,
-"w" + 3 "x" + 2 "y" + "z" == 3, 
 3 "w" + 2 "x" + 4 "y" + 5 "z" == 2, 
 8 "w" + 3 "x" + 5 "y" + 3 "z" == 4}

eqsfromMatrix[m1, 1] 

 {"w" + "y" + 3 "z" == "a",
  -"w" + 3 "x" + 2 "y" + "z" == "b", 
  3 "w" + 2 "x" + 4 "y" + 5 "z" == "c", 
  8 "w" + 3 "x" + 5 "y" + 3 "z" == "d"}

eqsfromMatrix[m1, 0] 

 {"w" + "y" + 3 "z" == 0, 
 -"w" + 3 "x" + 2 "y" + "z" == 0, 
 3 "w" + 2 "x" + 4 "y" + 5 "z" == 0, 
 8 "w" + 3 "x" + 5 "y" + 3 "z" == 0}