# Implication for linear dependence

I need to prove that if a set of vectors is linearly independent, then something happens. I want to show that if that set is linearly dependent, then that something doesn't happen.

So suppose $$V = \{v_1,v_2,...,v_n\}$$ are linearly dependent.

I know that two vectors are linearly dependent if one is a scalar multiple of the other and that the set $$V$$ above is linearly dependent if some linear combination of the vectors with nonzero scalars gives the zero vector.

Can I claim that if $$V$$ is linearly dependent, then there exist some $$v_i, v_j$$ such that $$v_i = cv_j$$? Does this follow from the above standard definition of linear independence for a set with more than 2 elements?

• No. One vector need not be a scalar of another in a linearly dependent set. Also, (linear dependence) $\implies$ (something) is not equivalent to showing (linear independence) $\implies$ (something). Commented Oct 13, 2022 at 3:35
• This almost never works outside of a set of size 2. For instance, take the set $S \subset \mathbb{R}^2$ defined by $S = \{\hat{e}_1, \hat{e}_2, \hat{e}_1 + \hat{e}_2\}$. You can't have a linearly independent set of size $3$ in $\mathbb{R}^2$ ($S$ is linearly dependent) and no pairs of vectors satisfy your listed property. Commented Oct 13, 2022 at 3:37

As @Ethan commented, take the example $$\begin{bmatrix}1\\0\end{bmatrix}$$, $$\begin{bmatrix}0\\1\end{bmatrix}$$, $$\begin{bmatrix}1\\1\end{bmatrix}$$. The third is the sum of both, but clearly none of them is a multiple of the other.