Step 1: GIVEN AND OBJECTIVE

We will first consider the given statement,

The transformation \(\displaystyle{T}:{R}_{{{n}}}\rightarrow\) R defined by T(x) =|x| is a linear transformation.

The objective is to determine the statement as true or false.

Step 2: EXPLANATION

As we know, \(\displaystyle{T}:{x}\rightarrow{\left|{x}\right|}\in{R}\) gives a norm on \(R_{n}\) and norm is not linear.

Since, norm satisfies the following,

\(\displaystyle{\left|{x}+{y}\right|}\leq{\mid}{x}{]}+{\left|{y}\right|}\) for all \(\displaystyle{x},{y}\in{R}_{{{n}}}\)

This implies that T is not linear.

Hence, the statement is False.