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

Step 1: Transformation
In mathematics, a transformation is a function f (usually with some geometrical underpinning) that maps a set X to itself, i.e f: ${\displaystyle X\to X}$
Examples:
(1) Linear transformation of vector spaces
(2) Geometric transformation
More generally, a transformation in mathematics means a mathematical function (i.e map or mapping). A transformation can be an invertible function from a set X to itself or X to another set Y.
Step 2: Linear transformation
A linear transformation is a transformation ${\displaystyle T:R^n\to R^m}$ satisfying
${\displaystyle T(u+v)=T\left(u\right)+T\left(v\right)}$
And
${\displaystyle T\left(cu\right)=cT\left(u\right)}$
For all vector u, v in ${\displaystyle R^n}$ and all scalars c.
Properties of the linear transformation
Let T${\displaystyle :R^n\to R^m}$ be a linear transformation. Then:
1) ${\displaystyle T\left(0\right)=0}$
2. For any vector ${\displaystyle v_1,v_1,v_2,v_3,\dots .v_k\in R^n}$ and scalars ${\displaystyle c_1,c_2,c_3,\dots \dots c_k}$
${\displaystyle T(c_1{v}_1+c_2{v}_2+c_3{v}_3+\dots \dots .+c_k{v}_k)=c_{1}T\left(v_1\right)+c_{2}T\left(v_2\right)+c_{3}T\left(v_3\right)+\dots \dots ..+c_{k}T\left(v_k\right)}$
Step 3
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 maintains 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.