Chapter 4 - Vector Spaces - 4.5 Linear Dependence and Linear Independence - Problems - Page 298: 53

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Assume that the set of vectors $\{v_1,v_2,...v_k\}$ is a spanning set for vector space $V$ and $v$ is a vector in $V$ . There exist scalars $a_1,a_2,...a_k$ such that: $\rightarrow v=a_1v_1+a_2v_2+...+a_kv_k$ $\rightarrow v-a_1v_1+a_2v_2+...+a_kv_k=0$ Hence, the set $\{v_1,v_2,...v_k\}$ is linearly independent.