# Math Help - Linear map and Linear independence

1. ## Linear map and Linear independence

Hi! I've been trying to figure this out, but I'm not sure how I'd prove it..

Let V, W be two vector spaces, and F: V-->W a linear map. Let w_1, ..., w_n be elements of W which are linearly independent, and let v_1, ..., v_n be elements of V such that F(v_i)=w_i for i = 1, ..., n. Show that v_1, ..., v_n are linearly independent.

$F(a_1v_1 + a_2v_2 + \cdots a_nv_n) = a_1F(v_1) + a_2F(v_2) + \cdots a_n F(v_n) = a_1w_1 + a_2w_2 + \cdots a_nw_n = 0 \Rightarrow a_1v_1 + a_2v_2 + \cdots a_nv_n = 0$.