The discussion centers on proving that the vector space V can be expressed as the direct sum of the ranges of two linear maps S and T, given that S + T = Iv and S ∘ T = Ov = T ∘ S. The key to the proof lies in utilizing the definition of the direct sum, which requires demonstrating that the intersection of the ranges X and Y is trivial and that their sum spans the entire space V. The participants emphasize the importance of confirming these conditions to establish V = X ⊕ Y.
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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?crypt50 said: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.