How do we show that $L$ is diagonalizable?

[Chris L T521]

Thanks to those who participated in last week's POTW! Here's this week's problem!

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Problem: Let $V$ be the vector space of continuous functions with basis $\{e^t,e^{-t}\}$. Let $L:V\rightarrow V$ be defined by $L(g(t)) = g^{\prime}(t)$ for $g(t)\in V$. Show that $L$ is diagonalizable.

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[Chris L T521]

This question was correctly answered by Siron and Sudharaka. You can Sudharaka's solution below:

Spoiler

Take any \(g(t)\in V\). Then,

\[g(t)=\alpha e^{t}+\beta e^{-t}\]

\[\Rightarrow g'(t)=\alpha e^{t}-\beta e^{-t}\]

Hence,

\[T\left(\begin{matrix} \alpha \\ \beta \end{matrix}\right)\rightarrow\left(\begin{matrix} \alpha \\ -\beta \end{matrix}\right)\]

Note that for each \(v=\left(\begin{matrix} \alpha \\ \beta \end{matrix}\right)\in V\)

\[\left(\begin{matrix} 1&0 \\ 0&-1 \end{matrix}\right)\left(\begin{matrix} \alpha \\ \beta \end{matrix}\right)=\left(\begin{matrix} \alpha \\ -\beta \end{matrix}\right)\]

The linear transformation \(T\) can be represented by a diagonal matrix with respect to the basis \(\{e^t,\,e^{-t}\}\). Hence \(T\) is diagonalizable.