Describing Transformations in Coordinates

In a previous post we discussed coordinate changes in linear algebra. In this post, we want to expand upon this by discussing descriptions of linear transformations in coordinates.

Using the notation from the previous article, if $X$ and $Y$ are spaces of objects, $\mathcal{A}$ is a bijective descriptive framework for objects in $X$ , and $\mathcal{B}$ is a bijective descriptive framework for objects in $Y$ , then $[x]_{\mathcal{A}}$ refers to the $\mathcal{A}$ description of $x \in X$ , and $[y]_{\mathcal{A}}$ refers to the $\mathcal{B}$ description of $y \in Y$ .

Now let’s suppose we are given a transformation $T: X \rightarrow Y$ . To describe this transformation with respect to the descriptive frameworks $\mathcal{A}$ and $\mathcal{B}$ , we want to define a function $_{\mathcal{B}}[T]_{\mathcal{A}}$ from the space of $\mathcal{A}$ -descriptions to the space of $\mathcal{B}$ -descriptions. The property this function should satisfy is that for any object $x \in X$ , it should take the $\mathcal{A}$ -description of $x$ to the $\mathcal{B}$ -description of $T(x)$ . In other words, for all $x \in X$ , $_{\mathcal{B}}[T]_{\mathcal{A}}[x]_{\mathcal{A}} = [T(x)]_{\mathcal{B}}$

Describing Transformations in Coordinates

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