Theorem. Let $T$ be a length-preserving linear transformation. Then either the matrix of $T$ is $\begin{pmatrix} \cos\theta & -sin\theta \\ sin\theta & cos\theta \end{pmatrix}$ for some number $\theta$ and then $T$ is rotation $R_\theta$, or else the matrix of $T$ is $\begin{pmatrix} \cos\theta & sin\theta \\ sin\theta & -cos\theta \end{pmatrix}$, and then $T$ is reflection through the line through the origin which forms an angle of $\theta /2$ with the positive $x-axis$.
My question is: I didn't understand the theorem that what is ''the the matrix of $T$ is $\begin{pmatrix} \cos\theta & -sin\theta \\ sin\theta & cos\theta \end{pmatrix}$ etc.'' mean? And how ca I use the therem in the isometry of the plane, can you give an axeample for using the theoerem?
This may not constitute a full answer to your question, but may provide some food for thought.
Suppose that $T$ is the $2 \times 2$ matrix of a linear transformation. The value of the determinant of $T$ tells several important things about the transformation.
If $R$ is a region of the plane and $R'$ is its image, then area of $R′ = | \det(T) | \cdot A(R)$, where $A(R)$ is the area of the region $R$.
(For example, if $\det(T) = 12$, every region is mapped to a region that is $12$ times as large.)
For example, all rotations above have a positive determinant, which you can find out for yourself.
For example, all reflections have a negative determinant, again you can check this above, for yourself.
For example, all projections have a zero determinant.