Define Transformation? Write down the properties of Linear transformation and rotational transformation.

In mathematics, a transformation is a f that maps a set X to itself, $i.e,f:X\to X.$
Examples:
(1). Linear transformation of vector space
(2) Geometric transformation
More generally, a transformation in mathematics means a mathematical function. A transformation can be an invertible function from a set X to itself or X to another set Y
A linear transformation is a transformation $T:R^n\to R^m\text{satisfying}T(u+v)=T(u)+T(v)\text{and}T(cu)=cT(u)$
for all vectors $u,v\in R^n$ and all scalars c.
Properties of the linear transformation:
Let $T:R^n\to R^m$ be a linear transformation. Then
$1.T(0)=0.$
2. For any vector $y_1,v_2,v_3,v_4,....v,\in R^n$
and scalars $c_1,c_2,...c_k,$
$T(c_1{v}_1,+c_2{v}_2,+...+c_k{v}_k)=c_{1}T(v_1)+c_{2}T(v_2)+...+c_{k}T(v_k)$
A rotational transformation is a transformation that turns a figure around a given point called the center of the rotation. The size and shape of the figure don’t change after rotation.
Properties of rotational transformation:
1. A rotation maintain the length of segments.
2. A rotation maintains the measure of angles.
3. A rotation maps a line to line, ray to ray, a segment to a segment, and an angle to an angle.