Please help me to find a linear transformation $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ that takes the line $2x+3y=6$ to the line $2x-3y=6$. I have found an affine transformation which is $T=(0,6)+\begin{pmatrix}1 & 0\\ 0 &-1 \end{pmatrix}$ but I can not find a linear transformation.
2026-04-01 09:32:50.1775035970
Find a linear transformation that takes a line to a line
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A generic linear transformation from $\mathbb{R}^2$ into $\mathbb{R}^2$ is of the type $f(x,y)=(ax+by,cx+dy)$. Both points $(3,0)$ and $(0,2)$ belong to the line $2x+3y=6$. So, pick $a$, $b$, $c$, and $d$ such that both $f(3,0)\bigl(=(3a,3c)\bigr)$ and $f(0,2)\bigl(=(2b,2d)\bigr)$ belong to the line $2x-3y=6$. What this means is that$$6a-9c=4b-6d=6.$$