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Exercise 8.C.20

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Proof. This is a bit obvious if you realize that

\dim G (λ, T) = \dim G (λ, T |_{V_{1}}) + \dots + G (λ, T |_{V_{m}})

for every $λ \in 𝔽$ . This is true because the subspaces on the right side of the equation are contained in $G (λ, T)$ and because when we write each $V_{j}$ as a direct sum of generalized eigenspaces of $T |_{V_{j}}$ in the equation

V = V_{1} \oplus \dots \oplus V_{m}

the only way the dimensions will fit is if $(2)$ is true. □

celiopassos

2017-10-06 00:00

Exercise 8.C.20

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