Exercise 2.C.9

Proof. The result clearly holds if $v_1+w,\dots ,v_m+w$ is linearly independent, so assume it is not. By Exercise 10 in section 2A, $w\in \mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(v_1,\dots <!--\; FUNCTION\; APPLICATION\; -->,v_m)$ we can write

for some unique scalars $a_1,\dots ,a_m\in ℝ$. Since $w$ is nonzero, we have $a_j\ne 0$ for at least one $j$. Removing $v_j+w$, we get a list of length $m-1$ which is linearly independent by previously cited exercise and the uniqueness of the scalars $a_i$. □