Mark each of the following statement true or false: The transformation T:

a2linetagadaW 2021-11-08 Answered
Mark each of the following statement true or false:
The transformation \(\displaystyle{T}:{R}_{{{n}}}\rightarrow{R}\) defined by T(x) = | x | is a linear transformation.

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Expert Answer

smallq9
Answered 2021-11-09 Author has 21332 answers

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.

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