# Span dimension, vector space dimension, spanning set

If the dimension of the span of a subset $$X$$ is equal to the dimension of the vector space $$V$$, is $$X$$ a spanning set of $$V$$?

• If the dimensions are finite then yes. The reason is that any large enough set of linearly independent vectors form a basis. – Zeekless Jun 1 at 9:00

In a space $$V$$ of dimension $$n$$ the only subspace $$X$$ of dimension $$n$$ is $$V$$ itself. Indeed, if there is a vector $$v$$ which is not in this subspace then the span of $$X\cup \{v\}$$ would have dimension greater than the dimension of $$V$$ which is impossible. Hence $$X$$ must be equal to $$V$$.