
theorem
for F being Field
for U,V being finite-dimensional VectSp of F
for B being Basis of U
for f being Function of B,V
for T being linear-transformation of U,V
holds T = canLinTrans f iff
      for u being Element of U st u in B holds T.u = f.u
proof
let F be Field, U,V be finite-dimensional VectSp of F;
let B be Basis of U, f be Function of B,V, T be linear-transformation of U,V;
H: dom f = B by FUNCT_2:def 1;
A: now assume T = canLinTrans f; then
   T|B = f by defcl;
   hence for u being Element of U st u in B holds T.u = f.u by FUNCT_1:49;
   end;
now assume C: for u being Element of U st u in B holds T.u = f.u;
  D: now let x be object;
     assume x in dom f; then
     x in B;
     hence f.x = T.x by C;
     end;
  dom T = the carrier of U & B c= the carrier of U by FUNCT_2:def 1; then
  dom f = dom T /\ B by H,XBOOLE_1:28; then
  T|B = f by D,FUNCT_1:46;
  hence T = canLinTrans f by defcl;
  end;
hence thesis by A;
end;
