Question 7

Note that a map T is linear iff both:

  1. \alpha T(v) = T(\alpha v) for all v \in V and \alpha \in \mathbb F
  2. T(u + v) = T(u) + T(v) for all u, v \in V

So if T is linear we have:

T(\alpha v_1 + \beta v_2) = T(\alpha v_1) + T(\beta v_2)

by property 1, and:

T(\alpha v_1) + T(\beta v_2) = \alpha T(v_1) + \beta T(v_2)

by property 2.

For the converse, suppose T(\alpha v_1 + \beta v_2) = \alpha T(v_1) + \beta T(v_2) for all v_1,v_2 and \alpha, \beta \in \mathbb F. We have T(\alpha v_1) = \alpha T(v_1) for all \alpha \in \mathbb F and v_1 \in V by setting \beta = 0. By setting \alpha = \beta = 1 we have T(v_1 + v_2) = T(v_1) + T(v_2) for all v_1, v_2 \in V. So T is linear.