Question

Show T is a linear transformation.

Answer 1

So the trace is a linear transformation if
1. Tr(A+B)=Tr(A)+Tr(B), where A,B are in your vector space
2. Tr(r*A)=r*Tr(A), where r is a real number.

Answer 2

Let's start with the first one. Formally, the trace is defined as \[Tr(A)=\sum_{i=1}^nA_{ii}.\]So\[
\begin{aligned}
Tr(A+B)&=\sum_{i=1}^n(A_{ii}+B_{ii})\\
&=\sum_{i=1}^nA_{ii}+\sum_{i=1}^nB_{ii}\\
&=Tr(A)+Tr(B)
\end{aligned}\]So the first condition is satisfied.

Do you think you could prove the second condition in the same manner?

Answer 3

That's what you need to prove. So hopefully it would be r*Tr(A).

Answer 4

You should ideally start with Tr(r*A) and end with r*Tr(A), but since you'll be using equalities, it won't matter too much.

Answer 5

That's basically it.

Answer 6

You're welcome.