For every distance preserving linear transformation $ \ S: \mathbb{R}^3 \to \mathbb{R}^3 \ $

Question

Select which are correct?

(1) For every distance preserving linear transformation $ \ S: \mathbb{R}^3 \to \mathbb{R}^3 \ $ , it is true that $$ \ || S(1,1,1)||=\sqrt 3 \ $$

(2) For every invertible linear transformation $ \ S: \mathbb{R}^3 \to \mathbb{R}^3 \ $ , it is true that $$ S(1,1,1) ||=\sqrt 3 $$

(3) For every linear transformation $ \ L : \mathbb{R}^3 \to \mathbb{R}^3 \ $ , it is true that $$ || L(2,1,-2)||=3 $$

Answer:

(1)

If $ S \ $ be distance preserving , then $ \ || S(x)||=||x || \ $

thus $ \ || S(1,1,1)||=||(1,1,1)||=\sqrt 3 \ $

The first option is true.

(3) It is false clearly.

what about the 2nd question ?

I need help .

Answer 1

Just a small hint: Try $S(x)=2x$.