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Is $\text{span} \space <1, 1, 1, 0>$ linearly independent?

We have that the only solution to $c \cdot <1, 1, 1, 0> $ is $c = 0$, so it is indeed linearly independent? But this is a single vector, does it still hold?

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    Is the question about the span of a single vector (which isn't a single vector) or a single vector? – 2017-01-09
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    I think that, strictly speaking, it’s not vectors that *are* linearly dependent or independent, but a set of vectors that *is* linearly dependent or independent. So a single vector composes a set that’s independent as long is the vector in question is nonzero. – 2017-01-09

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The span of a vector is not a vector, rather the set of linear combinations of that vector and thereby trivially linearly dependent. A vector $v \neq 0$ itself is always linearly independent since the equation $$\lambda v = 0$$ only has the solution $\lambda = 0$ (where $\lambda$ is a scalar).