- needs editing to find correct problem  --- 5. $ \begin{array}{l} \text{18. Suppose } V \text{ is finite-dimensional, with } \text{dim} V = n \geq 1 .\\\\ \text{ Prove that there exist } \text{one-dimensional subspaces } V_1, \ldots, V_n \text{ of } V \text{ such that} \\ V = V_1 \oplus \ldots \oplus V_n . \\\\ \text{SOLUTION: Let } v_1, \ldots, v_n \text{ be a basis of } V.\\\\ \text{ For each } k, \text{ let } V_k \text{ equal span} (v_k) ; \text{ in } \text{other words, } V_k = \{ a v_k : a \in \mathbb{F} \} . \\\text{ Because } v_1, \ldots, v_n \text{ is a basis of } V, \text{ each vector in } V \text{ can be written uniquely in the form} \\ a_1 v_1 + \ldots + a_n v_n , \\\\ \text{where } a_1, \ldots, a_n \in \mathbb{F} \text{ (see 2.28). By definition of direct sum, this means that} \\ V = V_1 \oplus \ldots \oplus V_n . \\ \text{Edward Frenkel} \end{array} $ --- Commentary: