0
$\begingroup$

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

|cite|improve this question
$\endgroup$
  • 3
    $\begingroup$ what would be your first step towards this??? suppose $v_1,v_2,\dots ,v_n$ is linearly dependent, then...... $\endgroup$ – user87543 Oct 3 '13 at 9:12
  • $\begingroup$ 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... $\endgroup$ – Sammy Black Oct 3 '13 at 9:19
  • $\begingroup$ ah i see, it would give the linear map a combination therefore giving a contradiction. thanks :) $\endgroup$ – Tessa Danger Bamkin Oct 3 '13 at 9:21
4
$\begingroup$

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?

|cite|improve this answer
$\endgroup$
2
$\begingroup$

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

|cite|improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.