Chapter 6 - Linear Transformations - 6.3 The Kernel and Range of a Linear Transformation - Problems - Page 407: 24

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For every $v \in V$ there exist scalars $a_1,a_2,...,a_k$ such as: $v=a_1v_1+a_2v_2+...+a_kv_k$ We obtain: $T(v)=T(a_1v_1+a_2v_2+...+a_kv_k)\\ =a_1T(v_1)+a_2T(v_2)+...+a_kT(v_k)$ Since $a_1T(v_1)=a_2T(v_2)=...=a_kT(v_k)$ $=a_1.0+a_2.0+...+a_k.0\\ =0$ with $v \in V$ Hence, $T$ is a zero transformation.