Some Properties of trace of linear maps.

Question

I need help have to prove some properties of trace

Let S and T be linear maps on an Euclidean space V,

1. Show that $$\operatorname {trace} (ST)=\operatorname {trace} (TS)$$
2. Prove or disprove that $$\operatorname {trace} (ST)=\operatorname {trace} (T)\cdot\operatorname {trace} (S)$$
3. If for all linear map K $\operatorname {trace} (TK)= 0$ then show that, $T=0$

I was able to solve the first question using the fact that for given matrices A and B we have,

$$(AB)_{i,j} = \sum_{k}a_{ik}b_{kj}$$

Hence $$\operatorname {trace} (AB)= \sum_{i}(AB)_{i,i} = \sum_{i}\sum_{k}a_{ik}b_{ki}=\sum_{k}\sum_{i}b_{ki}a_{ik} = \sum_{i}(BA)_{i,i}=\operatorname {trace} (AB) $$ is this correct? can anyone help with questions 2 and 3?

Answer 1

1. All good.
2. What about $S=T=\operatorname{Id}$? Or any random matrices provided you are not cursed with bad luck.
3. Hint: what is $\operatorname{Tr}(MM^T)$ for a given matrix $M$?