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

It will be a linear transformation if it satisfies 3 conditions:

\[1) T(A_{1} + A _{2}) = T(A _{1}) + T(A _{2}) \]\[2) T(\alpha A ) = \alpha T(A)\]\[3) T(O _{a}) = O _{b}\] Where "alpha" is a constant, "Oa" is the zero of the domain and "Ob" is the zero of the codomain (in this example, they're the same zeros).

In 1), we have: \[T(A _{1}+A _{2}) = (A _{1}+A _{2})B - B(A _{1}+A _{2}) = A _{1}B + A _{2}B - BA _{1}-BA _{2} = \]\[[A _{1}B - BA _{1}] + [A _{2}B - BA _{2}] = T(A _{1}) + T(A _{2})\]

For the second condition: \[T(\alpha A) = (\alpha A)B - B(\alpha A) = \alpha AB - \alpha BA = \alpha (AB - BA) = \alpha T(A)\]

For the last one: \[T(O) = OB - BO = O - O = O\]

Therefore, it's linear.

Ps.: You could have done the first two conditions at once, doing: \[T(\alpha A _{1} + \beta A _{2}) = \alpha T(A _{1}) + \beta T(A _{2}) \] I just did that way to be more clear!

Hope it helped, and sorry for any grammar mistakes!