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

crypt50

New member
Jun 29, 2013
21
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
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?