You seem to confuse transformations with conversion of units. A transformation is a purely mathematical concept and as such has no notion of (physical) units. Physics gives rise to the concept of units which are used to describe physical quantities.

Coming back to your example, let's consider a one-dimensional coordinate system and a pen therein. The pen's head is located at $\vec{x}_{head}=0$ and its tail is located at $\vec{x}_{tail}=L$. These two positions are (one-dimensional) vector quantities while $L$, the pen's length, is a scalar quantity. Now consider a reflection of the coordinate system about $\vec{x}=0$. The tail position is going to change to $\vec{x}'_{tail} = -L$ but the pen's length is still $L$, i.e. it is invariant under such transformations (you could define the length as $L = ||\vec{x}_{tail} - \vec{x}_{head}||$).

Now you brought in the concept of (physical) units but, as mentioned above, this is distinct from (mathematical) transformations. While you wrote "transformation law" the appropriate term would be "conversion law" and this also emphasizes the distinction between these two concepts.

You seem to confuse transformations with conversion of units. A transformation is a purely mathematical concept and as such has no notion of (physical) units. Physics gives rise to the concept of units which are used to describe physical quantities.

Coming back to your example, let's consider a one-dimensional coordinate system and a pen therein. The pen's head is located at $\vec{x}_{head}=0$ and its tail is located at $\vec{x}_{tail} = t$. These two positions are (one-dimensional) vector quantities while the pen's length is a scalar quantity, defined by $L = ||\vec{x}_{tail} - \vec{x}_{head}||$ where $||\cdot||$ is an appropriate metric with which the vector space is equipped (usually Euclidean metric in physics context but Minkowski metric is used as well). Now consider a reflection of the coordinate system about $\vec{x}=0$. The tail position is going to change to $\vec{x}'_{tail} = -t$ but the pen's length is still $L$, i.e. it is invariant under such transformations.

Now you brought in the concept of (physical) units but, as mentioned above, this is distinct from (mathematical) transformations. While you wrote "transformation law" the appropriate term would be "conversion law" and this also emphasizes the distinction between these two concepts. As you can observe from the previous paragraph, no notion of a unit was necessary to describe the transformation (reflection) of the coordinate system. Units on the other hand determine the value of $t$ (the pen's tail position) that we are going to obtain by our measurement. They ensure that you and I, if we measure the same pen, arrive at the same value for that position. A conversion of unit means that we change to a different ruler and hence obtain a different number (magnitude) for the position but this is all valid and compatible as long as we report this number together with its unit of measurement and we know the relevant conversion rules.

You seem to confuse transformations with conversion of units. A transformation is a purely mathematical concept and as such has no notion of (physical) units. Physics gives rise to the concept of units which are used to describe physical quantities.

Coming back to your example, let's consider a one-dimensional coordinate system and a pen therein. The pen's head is located at $x=0$ and its tail is located at $x = L$. These two positions are (one-dimensional) vector quantities while $L$, the pen's length, is a scalar quantity. Now consider a reflection of the coordinate system about $x=0$. The tail position is going to change to $x' = -L$ but the pen's length is still $L$, i.e. it is invariant under such transformations (you could define the length as $L = |x_{tail} - x_{head}|$).

Now you brought in the concept of (physical) units but, as mentioned above, this is distinct from (mathematical) transformations. While you wrote "transformation law" the appropriate term would be "conversion law" and this also emphasizes the distinction between these two concepts.

You seem to confuse transformations with conversion of units. A transformation is a purely mathematical concept and as such has no notion of (physical) units. Physics gives rise to the concept of units which are used to describe physical quantities.

Coming back to your example, let's consider a one-dimensional coordinate system and a pen therein. The pen's head is located at $\vec{x}_{head}=0$ and its tail is located at $\vec{x}_{tail}=L$. These two positions are (one-dimensional) vector quantities while $L$, the pen's length, is a scalar quantity. Now consider a reflection of the coordinate system about $\vec{x}=0$. The tail position is going to change to $\vec{x}'_{tail} = -L$ but the pen's length is still $L$, i.e. it is invariant under such transformations (you could define the length as $L = ||\vec{x}_{tail} - \vec{x}_{head}||$).

Now you brought in the concept of (physical) units but, as mentioned above, this is distinct from (mathematical) transformations. While you wrote "transformation law" the appropriate term would be "conversion law" and this also emphasizes the distinction between these two concepts.