# Intuition - Zero Vector, Its Existence in Any Set, and Linear Dependence [GStrang, P169]

$I.$ The zero vector is linearly dependent.
$II.$ Any set containing the zero vector must be linearly dependent.

I only apprehend the truths of I and II above from the definition of linear independence : http://math.berkeley.edu/~mjv/lindep0.pdf
www.math.ku.edu/~lerner/LAnotes/Chapter11.pdf

Howbeit, I'm not perceiving the intuition behind $I$ and $II$. For instance, in this first set of 2 pictures, the $\color{red}{u}$ $\in \mathbb{R}^2$ originates from $\color{#1E90FF}{0},$ and thus makes contact/"intersects" with $\color{#1E90FF}{0}$. So how and why isn't it dependent (on $\color{#1E90FF}{0})$?

Beyond geometric intuition(s) and this interpretation, what are other intuitions for $I$ and $II$?

This page precedes basis, dimension, Orthogonality, Determinants, Eigenvalues and eigenvectors, and linear transformations, so please omit these concepts in responses.

-
There is a linear combination of the set $0$ which gives the $0$ vector. Geometrically, when you are given a single vector $v_1$, you expect its span to be $1$-dimensional. However, the span of $0$ is $0$-dimensional. – Dustan Levenstein Nov 22 '13 at 5:41
If the zero vector was linearly independent, then $\vec{0}=\alpha\,\vec{0}$ would imply $\alpha=0$. – David H Nov 22 '13 at 5:59