nimble island
Hello, I am studying affine geometry, particularly the theorem attached as an image. I asked ChatGPT to give me a numerical example of this (since the book doesn't include one). I would like to know if it is correct. It seems strange to me that it doesn't consider the vector (\overrightarrow{X_2X_3}) and (\overrightarrow{Y_2Y_3}). Also, in step 2, I don't understand how it was converted into matrix form. :c
Given the vertices:
\begin{itemize}
\item (X_1(1,1)), (X_2(2,3)), (X_3(4,2))
\item (Y_1(2,2)), (Y_2(3,5)), (Y_3(6,4))
\end{itemize}
\subsubsection{Step 1: Calculate the vectors of the triangles}
For the triangle (X_1X_2X_3):
[
\overrightarrow{X_1X_2} = \begin{pmatrix} 2 - 1 \ 3 - 1 \end{pmatrix} = \begin{pmatrix} 1 \ 2 \end{pmatrix}, \quad \overrightarrow{X_1X_3} = \begin{pmatrix} 4 - 1 \ 2 - 1 \end{pmatrix} = \begin{pmatrix} 3 \ 1 \end{pmatrix}
]
For the triangle (Y_1Y_2Y_3):
[
\overrightarrow{Y_1Y_2} = \begin{pmatrix} 3 - 2 \ 5 - 2 \end{pmatrix} = \begin{pmatrix} 1 \ 3 \end{pmatrix}, \quad \overrightarrow{Y_1Y_3} = \begin{pmatrix} 6 - 2 \ 4 - 2 \end{pmatrix} = \begin{pmatrix} 4 \ 2 \end{pmatrix}
]
\subsubsection{Step 2: Set up the system for the matrix (L)}
We seek the matrix (L) such that:
[
L \begin{pmatrix} 1 \ 2 \end{pmatrix} = \begin{pmatrix} 1 \ 3 \end{pmatrix}, \quad L \begin{pmatrix} 3 \ 1 \end{pmatrix} = \begin{pmatrix} 4 \ 2 \end{pmatrix}
]
This can be written in matrix form as:
[
L \begin{pmatrix} 1 & 3 \ 2 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 4 \ 3 & 2 \end{pmatrix}
]
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