# This is an example of how to use maple to find the representation matrix
# B for T(x)=Ax relative to a new basis u1, u2, u3. The columns of A are
# called v1,v2,v3. The vectors in the new basis are defined by
#   u1 = v1 + 2 v3
#   u2 = v1 + 3 v2
#   u3 = v2 + 4 v3
#
> with(linalg):# Bretscher 3E, p142
> v1:=vector([1,0,0]);v2:=vector([1,1,0]);v3:=vector([1,0,1]);
> A:=augment(v1,v2,v3);
> P:=augment(v1+2*v3,v1+3*v2,v2+4*v3);
> # Solve the equation d1 u1 + d2 u2 + d3 u3 = T(u1) for d1, d2, d3.
> # Then d:=vector([d1,d2,d3]) == col(B,1). Or, solve P d = T(u1) for d.
> # The answer is d := inverse(P) times T(u1).
> u1:=col(P,1);b1:=evalm(inverse(P) &* A &* u1);
> u2:=col(P,2);b2:=evalm(inverse(P) &* A &* u2);
> u3:=col(P,3);b3:=evalm(inverse(P) &* A &* u3);
> B:=augment(b1,b2,b3);