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Exercise 2.2.16

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As the process in the proof of Dimension Theorem, we may pick the same basis $β = {v_{1}, v_{2}, \dots v_{n}}$ for $V$ . Write $u_{k + 1} = T (v_{k + 1}$ . It has been proven that ${u_{k + 1}, u_{k + 2}, \dots, u_{n}}$ is a basis for $R (T)$ . Since dim $(V) =$ dim $(W)$ , we can extend to a basis $γ = {u_{1}, u_{2}, \dots, u_{n}}$ for $W$ . Thus we have

{[T]}_{β}^{γ} = (\begin{matrix} O & O \\ O & I_{n - k} \end{matrix})

is a diagonal matrix.

jephianlin

2011-06-27 00:00

Exercise 2.2.16

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