What is the generalization of Lie group of transformation?
I found $a_1x+a_2$ and $(a_1x+a_2)/(a_3x+a_4)$ are also called Lie group of transformation!! It contradicts with what we learn about the Type A, Type B, Type C ...
any book or web link that lists all Lie groups of transformation or family of Lie group of transformation for reference would be nice.
Any one using the following Maple code to demonstrate the steps from matrix to these equations?
Maple Code
restart; with(LinearAlgebra): with(linalg): TypeB := proc(n) m := Matrix(n); for i from 1 to n do for j from 1 to n do if i = j then if i+1 < n then m[i+1,j] := -1; end if; if i+1 = n then m[i+1,j] := -2; end if; m[i,j] := 2; if j+1 <= n then m[i,j+1] := -1; end if; end if; od; od; return m; end proc; TypeC := proc(n) m := Matrix(n); for i from 1 to n do for j from 1 to n do if i = j then if i+1 <= n then m[i+1,j] := -1; end if; m[i,j] := 2; if j+1 <= n then m[i,j+1] := -1; end if; if j+1 = n then m[i,j+1] := -2; end if; end if; od; od; return m; end proc; TypeD := proc(n) m := Matrix(n); for i from 1 to n do for j from 1 to n do if i = j then if i+1 <= n then m[i+1,j] := -1; end if; if i+2 = n then m[i+2,j] := -1; end if; if i+1 = n then m[i+1,j] := 0; end if; m[i,j] := 2; if j+1 <= n then m[i,j+1] := -1; end if; if j+2 = n then m[i,j+2] := -1; end if; if j+1 = n then m[i,j+1] := 0; end if; end if; od; od; return m; end proc; TypeA := proc(n) m := Matrix(n); for i from 1 to n do for j from 1 to n do if i = j then if i+1 <= n then m[i+1,j] := -1; end if; if i-2 = 0 then m[i+2,j] := -1; end if; m[i,j] := 2; if j+1 <= n then m[i,j+1] := -1; end if; end if; od; od; return m; end proc; TypeF4 := proc(n) m := Matrix(4); for i from 1 to 4 do for j from 1 to 4 do if i = j then if i+1 <= 4 then m[i+1,j] := -1; end if; m[i,j] := 2; if j+1 <= 4 then m[i,j+1] := -1; end if; end if; od; od; m[3,2] := -2; return m; end proc; TypeG2 := Matrix([[2,-1],[-3,2]]);