Question

I don't understand how to do this problem in my homework: Show that the given mapping is a linear transformation T: Mn(R) -> Mn(R) defined by T(A)= AB-BA where B is a fixed n x n is matrix. Help please!

Answer 1

Show that T(A+B)=T(A)+T(B)

Answer 2

I understand I have to do that but I don't know how the process works..

Answer 3

LHS=T(A+C)=(A+C)B-B(A+C)=AB+CB-BA-BC
=(AB-BA)+(CB-BC)
=T(A)+T(C)..........gET tHAT

Answer 4

you should also show that T(cA)=cT(A)

Answer 5

I was about to ask about that part, how would that be done?

Answer 6

Well, T(cA)=(cA)B-B(cA)=c(AB)-c(BA)=c(AB-BA)=cT(A).