Assume that S and T are linear maps from the vector space V to itself.

[crypt50]

Assume also that S + T = Iv and that S ∘ T = Ov = T ∘ S. Prove that V = X ⊕ Y where
X = range(S) and Y = range(T). I don't understand how to go about it, please help.

 

[Opalg]

Assume also that S + T = Iv and that S ∘ T = Ov = T ∘ S. Prove that V = X ⊕ Y where
X = range(S) and Y = range(T). I don't understand how to go about it, please help.

You need to use the definition of the direct sum $X\oplus Y$. The question tells you which subspaces to use for $X$ and $Y$, so what do you have to check in order to show that the definition of $V = X\oplus Y$ is satisfied?