The Matrix Representation Of A Transformation In Different Basis Are Similar - Neutrinos are the coolest particles

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Let $V$ be a finite dimensional vector space with basis $B=\{x_1, \dots, x_n\}$ and $C=\{y_1, \dots, y_n\}$ and $T:V\to V$. Then $[T]_b$ and $[T]_C$ are similar. Specifically, for $$\begin{align*} P=\begin{bmatrix}[y_1]_B & \dots & [y_n]_B\end{bmatrix}, \end{align*}$$ $[T]_C=P^{-1}[T]_BP$

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Let $V$ be a finite dimensional vector space with basis $B=\{x_1, \dots, x_n\}$ and $C=\{y_1, \dots, y_n\}$ and $T:V\to V$. Then $[T]_b$ and $[T]_C$ are similar. Specifically, for $$\begin{align*} P=\begin{bmatrix}[y_1]_B & \dots & [y_n]_B\end{bmatrix}, \end{align*}$$ $[T]_C=P^{-1}[T]_BP$

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concepts

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FullPage

result

concepts

hypothesis

implications

proof

Let $V$ be a finite dimensional vector space with basis $B=\{x_1, \dots, x_n\}$ and $C=\{y_1, \dots, y_n\}$ and $T:V\to V$. Then $[T]_b$ and $[T]_C$ are similar. Specifically, for $$\begin{align*} P=\begin{bmatrix}[y_1]_B & \dots & [y_n]_B\end{bmatrix}, \end{align*}$$ $[T]_C=P^{-1}[T]_BP$

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Let $V$ be a finite dimensional vector space with basis $B=\{x_1, \dots, x_n\}$ and $C=\{y_1, \dots, y_n\}$ and $T:V\to V$. Then $[T]_b$ and $[T]_C$ are similar. Specifically, for $$\begin{align*} P=\begin{bmatrix}[y_1]_B & \dots & [y_n]_B\end{bmatrix}, \end{align*}$$ $[T]_C=P^{-1}[T]_BP$

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