Why $\{0\}$ is linearly dependent?

Question

Consider the case $x\in V, x\not=0_v$, then $S=\{x\}$ is linearly independent. I read this as "In the set S, only x itself can express x.". Ok, fine, it's by my intuition. But then I think about $\{0\}$, by definition it's linearly dependent. For me it's unfair since "$0$ itself can express itself" means "someone else can also express $0$".

It seems like the definition for the case of $0$ should be defined, but luckily in this definition of linearly dependent just fits, so we don't have to.

Why $\{0\}$ is linearly dependent?

Answer 1

Quoting the first line of the Wikipedia page on linear independence:

[...A] set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the others[.]

$\{0\}$ is a linearly dependent set. $0$ can be written as a linear combination of the other vectors in the set. Namely, the empty linear combination. $$\sum_{v\in\varnothing}k_v v = 0.$$

Answer 2

Talking about intuition: a good geometric intuition, at least for finite-dimensional vector spaces, is to say that a set of vectors is linearly independent iff the number of the vectors equals the dimension of their span.

As $\{0\}$ contains one vector but the dimension of its span is zero, $\{0\}$ is linearly dependent.