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I'm doubting myself a little bit on my logic when answering this question:

Suppose $T$ is a linear map, and $v_1,...,v_k$ are vectors. If $T(v_1), ..., T(v_k)$ are dependent, need $v_1, ..., v_k$ be dependent?

My reasoning is that yes, they do need to be dependent since: If $T(v_1), ..., T(v_k)$ are dependent, then there are some not-all-zero real numbers $c_1,...,c_k$ for which the equation $$c_1T(v_1) + ... + c_kT(v_k) = 0$$ holds. Suppose $F$ is a linear map that is the reverse of $T$. Then: $$c_1F(T(v_1)) + ... + c_kF(T(v_k)) = 0$$ which simplifies to: $$c_1v_1 + ... + c_kv_k = 0$$ Thus, $v_1,...,v_k$ are also linearly dependent.

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    $\begingroup$ Not all linear maps have "reverses". $\endgroup$ – kimchi lover Sep 12 '17 at 1:39
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No.

Take $T : \mathbb{R}^2 \to \mathbb{R}^2$ such that $T(a,b)=(a,a)$

If $e_1=(1,0)$ and $e_2=(0,1)$ then $T(e_1),T(e_2)$ are linearly dependent but $e_1,e_2$ are not.

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If map is injective then your logic is correct.injective implies karnal trivial.

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