The Span of a Finite Family is the Smallest Subspace Containing It

Let $K$ be a field, let $V$ be a vector space over $K$, let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, let $v\in V^{n}$ be an $n$-tuple in $V$ with components $v_{k}$, and let $\operatorname{span}(v)$ be its span. Then the following hold.

1. (Subspace) $\operatorname{span}(v)$ is a linear subspace of $V$, and $v_{j}\in\operatorname{span}(v)$ for every $j\in[n]$.

2. (Spanning) By claim 1 and claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product, the set $\operatorname{span}(v)$ is a vector space over $K$ under the operations of $V$ restricted to it. Let $v^{\ast}\in\operatorname{span}(v)^{n}$ be the $n$-tuple in $\operatorname{span}(v)$ with the same components $v_{k}$. Then $v^{\ast}$ spans that vector space.

3. (Smallest such subspace) If $W$ is a linear subspace of $V$ with $v_{k}\in W$ for every $k\in[n]$, then $\operatorname{span}(v)\subseteq W$.