Exercise 9.A.4

Proof. Suppose $v_1,\dots ,v_m$ spans $V_{ℂ}$. Let $v\in V$. Then $v+i0\in V_{ℂ}$ and we can write

for some ${\lambda }_1,\dots ,{\lambda }_m\in ℂ$. The equation above implies that

Therefore $v\in \mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(v_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,v_m)$. Hence $v_1,\dots ,v_m$ spans $V$.

Conversely, suppose $v_1,\dots ,v_m$ spans $V$. Then we can reduce this list to a basis of $V$. But a basis of $V$ is also a basis $V_{ℂ}$. Therefore we can reduce $v_1,\dots ,v_m$ to a basis of $V_{ℂ}$ and this implies that it spans $V_{ℂ}$. □