When talking about linear independence of a family of vectors $(x_{1}, x_{2}, ... , x_{t})$, where family of vectors allows repetition and order matters, it's not set of vectors, we have below proposition:
$**Proposition**$
Let $V$ be a vector space, let $(x_{1}, x_{2}, ... , x_{t})$ be a family of vectors in $V$, and suppose $x_{1} \neq 0$, then $(x_{1}, x_{2}, ... , x_{t})$ is linearly independent if and only if for each integer, $j, j = 2, ..., s, x_{j} \notin \langle x_{1}, x_{j-1}\rangle$
My question is that why does this proposition needs the assumption that $x_{1} \neq 0$, and it even emphasizes on this assumption? What if $x_{1} = 0$, what effect would it be?