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

Ayaana Buck 2020-12-03 Answered
Define Transformation? Write down the properties of Linear transformation and rotational transformation.
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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
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
$1.T\left(0\right)=0.$
2. For any vector ${y}_{1},{v}_{2},{v}_{3},{v}_{4},....v,\in {R}^{n}$
and scalars ${c}_{1},{c}_{2},...{c}_{k},$
$T\left({c}_{1}{v}_{1},+{c}_{2}{v}_{2},+...+{c}_{k}{v}_{k}\right)={c}_{1}T\left({v}_{1}\right)+{c}_{2}T\left({v}_{2}\right)+...+{c}_{k}T\left({v}_{k}\right)$
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.