Question-22

Let \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}\) be a function. Select all linear transformations.

• \(T(v) = ||v||\)
• \(T(v) = v_1 + v_2 + v_3\), where \(v = (v_1, v_2, v_3)\)
• \(T(v) = \text{smallest component of } v\)
• \(T(v) = 0\)

Answer

• \(T(v) = ||v||\)
• \(T(v) = v_1 + v_2 + v_3\), where \(v = (v_1, v_2, v_3)\)
• \(T(v) = \text{smallest component of } v\)
• \(T(v) = 0\)

Solution

The correct options are quite easy to see. Let us see why the incorrect options are not linear transformations.

Option-1

This doesn’t satisfy \(T(cv) = cT(v)\). For example, \(T(-v) = ||v||\) and not \(-||v||\).

Option-3

This doesn’t satisfy \(T(cv) = cT(v)\). For example, \(v = (1, 2, 3)\). Here \(T(v) = 1\). But \(T(-v) = -3\) and not \(-1\).