Linear Algebra: Equivalence of Linear Transformations

[Fringhe]

Homework Statement

1) two linear transformations B and C are equivalent iff there exist invertible linear transformations P and Q such that PB=CQ
2) if A and B are equivalent then so are A' and B' in dual space
3) Do there exist linear transformations A and B such that A and B are equivalent but A^2 and B^2 are not?
4) Does there exist a linear transformation A such that A is equivalent to a scalar a but A is not equal to a?

The Attempt at a Solution

I really don't know where to start. I know that if two l.ts. A and B are equivalent then (AB)^-1 = B^-1A^-1. But that's where I am now.

 

[Fringhe]

Ok for the first question, two lts B and C are equivalent iff there exist lts E and F such that
B = E^-1 C F
Now let E = P and let F=Q, we have
B= P^-1 C Q or PB = CQ so this means that the lts P and Q must be invertible?

 

[Tedjn]

Can you please repeat your definition for equivalence between A and B? I'm not sure I follow.