0

Let $V$ and $W$ be vector spaces over $F$ , let $f\in \operatorname[Hom]_F(V,W)$ and let $ \vec v_1 \ldots \vec v_k \in V$ . Prove that if the set ${f(\vec v_1), \ldots , f(\vec v_k )}$ is linearly independent then $ \vec v_1 \ldots \vec v_k $ must be linearly independent.

not sure how to go about this one

Martin Sleziak
  • 53,687
Tessa Danger Bamkin
  • 283
  • 3
    what would be your first step towards this??? suppose $v_1,v_2,\dots ,v_n$ is linearly dependent, then...... –  Oct 03 '13 at 09:12
  • To prove that the set $\vec{v}_1, \dots, \vec{v}_k$ is linearly independent, suppose that $a_1 \vec{v}_1 + \dots, a_k \vec{v}_k = \vec{0}$ for some coefficients $a_1, \dots, a_k \in F$. Now, apply the linear map $f$ and use the given fact... – Sammy Black Oct 03 '13 at 09:19
  • ah i see, it would give the linear map a combination therefore giving a contradiction. thanks :) – Tessa Danger Bamkin Oct 03 '13 at 09:21

2 Answers2

4

Suppose $a_1,...,a_k\in F$ such that $a_1\vec v_1+\cdots+a_k\vec v_k=\vec 0.$ What can you do with $f(a_1\vec v_1+\cdots+a_k\vec v_k)=f(\vec 0),$ given that $f$ is a homomorphism?

Cameron Buie
  • 102,994
3

$c_1v_1+\dots+c_nv_n=0$ now apply $f$

Myshkin
  • 35,974
  • 27
  • 154
  • 332