Here's the question:

Let $S$ be a basis for an $n$-dimensional vector space $V$. Show that $\{ v_1, v_2, \dots , v_r \} \subseteq V$ is linearly independent if and only if the coordinate vectors $\{ (v_1)_S, (v_2)_S, \dots , (v_r)_S \} \subseteq \mathbb{R}^n$ is linearly independent.

I saw the answers to a similar (well, essentially the same) question here: Proving linear independence of a basis from coordinate vectors

However, at this point in the text I am using, the facts listed in the answers have not been introduced. And technically, we don't know that we can obtain the set $\{ (v_1)_S, (v_2)_S, \dots , (v_r)_S \}$ via a change of basis.

I'd like to prove this from more elementary means, using the definition of coordinate vectors. This is just the vector of coefficients when each vector in the original set is expressed in terms of the basis vectors.

Approach: I want to prove that

$k_1v_1 + \cdots + k_r v_r = 0 \Longrightarrow k_i=0~ \forall i$ if and only if $l_1(v_1)_S + \cdots + l_r(v_r)_S = 0 \Longrightarrow l_i = 0 ~ \forall i$.

However, if I express the left side as $S$-vectors, the vectors on the right side would be scalars on the left side. How do I translate without invoking a change of basis?

Thanks for your insights, I only need a nudge here.


Hint: proceed by contradiction and use the fact that the coordinate mapping is linear.

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