Problem-20

linear maps

Question

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

  1. \(T(v) = v/||v||\)

  2. \(T(v) = v_1 + v_2 + v_3\), where \(v = (v_1, v_2, v_3)\)

  3. \(T(v) = \text{smallest component of } v\)

  4. \(T(v) = 0\)

Solution

Options (b) and (d) are correct.

  • Option (a) is clearly wrong as \(T(0)\) is not defined. That is, the domain doesn’t even contain \(0\). On the other hand, a linear transformation should take the \(0\) vector to \(0\).

  • Option-(b) is correct. It can be verified that:

    • \(T((x_1, x_2, x_3) + (y_1, y_2, y_3)) = T(x_1, x_2, x_3) + T(y_1, y_2, y_3)\)
    • \(T(c(x, y, z)) = cT(x, y, z)\)

  • Option (c) is wrong. Here is one example where the linearity is broken:

    • \(T((-1, 0, 0) + (1, 0, -1)) = T(0, 0, -1) = -1\). But \(T(-1, 0, 0) + T(1, 0, -1) = -2\).

  • Option-(d) is correct. It is the zero-transformation.