Exercise 2A Problem 9

Problem

Prove or give counterexample: If $v_1, \dots, v_m$ and $w_1, \dots, w_m$ are linearly independent lists of vectors in $V$, then $v_1 + w_1, v_2 + w_2, \dots, v_m + w_m$ is linearly independent.

Solution

Let $w_1 = -v_1, w_2 = -v_2, \dots, w_m = -v_m$, then $v_1, v_2, \dots, v_m$ and $w_1, w_2, \dots, w_m$ are both linearly independent lists, but $v_1 + w_1, v_2 + w_2, \dots, v_m + w_m$ is just a list of zeros and is therefore clearly linearly dependent.