$T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $ Let V a vector space and $T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $ then $V = H_1 \bigoplus H_2$
I stuck in this problem please some help.

Hint: Just assume for the moment that you can write $v=h_1+h_2$ with $h_i\in H_i$. Can you express $Tv$ in terms of $h_1, h_2$? Can you then obtain $h_1,h_2$ from $v$ and $Tv$? 
