Linearly independent vectors each subtracted by a linear combination of them are linearly dependent if coefficients add up to $1$ The task
I've been given the following problem:

Let $v_1, \ldots, v_n$ linearly independent vectors in an $\mathbb{F}$-vector space $V$ and $u = \lambda_1 v_1 + \ldots + \lambda_n v_n$ a linear combination of those vectors.
Prove that
$$v_1 - u, \ldots, v_n - u \text{ are linearly dependent} \Leftrightarrow \lambda_1 + \ldots + \lambda_n = 1 $$

Unfortunately, I have some trouble with this question. What steps would need to be taken in order to prove this? One hint might be the fact that if a set of vectors is linearly independent, it means any linear combination of them has only one clear set of coefficients (and the other way around).
EDIT: After angryavian's answer, I've understood how to prove the first direction. Unfortunately, the other direction still leaves me quite puzzled. A hint would be greatly appreciated.

If you consider the matrix having as columns the coordinates of the new vectors with respect to the given linearly independent set (everything takes place in the subspace of which $\{v_1,\dots,v_n\}$ is a base):
\begin{bmatrix}
1-\lambda_1 & -\lambda_1 & \dots & -\lambda_1 \\
-\lambda_2 & 1-\lambda_2 & \dots & -\lambda_2 \\
\vdots & \vdots & \ddots & \vdots \\
-\lambda_n & -\lambda_n & \dots & 1-\lambda_n
\end{bmatrix}
you see that this matrix is
$$
I-
\begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n \end{bmatrix}
\begin{bmatrix} 1 & 1 & \dots & 1 \end{bmatrix}
$$
which is not invertible (meaning that $\{v_1-u,\dots,v_n-u\}$ is linearly dependent) if and only if $1$ is an eigenvalue of
$$
A=
\begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n \end{bmatrix}
\begin{bmatrix} 1 & 1 & \dots & 1 \end{bmatrix}
$$
This is a matrix with rank at most one, so it has at most one nonzero eigenvalue, namely
$$
\begin{bmatrix} 1 & 1 & \dots & 1 \end{bmatrix}
\begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n \end{bmatrix}
=\lambda_1+\lambda_2+\dots+\lambda_n
$$
