Let $V$ be real $n$-dimensional vector space, and $T:V\to V$ is a linear map satisfying the condition $T^2(v)=-v$ for all $v \in V$. Then,
- Show that $n$ is an even integer.
- Use $T$ to make $V$ into a complex vector space such that the multiplication by complex numbers extends the multiplication by reals.
- Show that, with respect to the complex vector space structure on $V$ obtained in 2, $T:V\to V $ is a complex linear function.
This problem is bugging me for a while. And I have a few questions about it. I did no. 1 using the concept of minimal polynomials. [Another nice proof can be found here.] But the real troubles are question no. 2 and 3. The whole statement of Q.2 looks very vague to me. (For instance, I have doubts that, if I declare a real vector space to be a complex one, how can it be same as the previous one?) The follow up question has an equally dubious statement.
I would be glad if somebody takes the time to clarify what these two statements actually mean and exactly what I have to prove. Thank you.
[Source: This question can be found here (Question 25b).]