Basis Extension Theorem I know that using basis extension theorem we can extend a set of LI vector to the basis of vector space, but how we actually do it?
I have only theoretically used this theorem in proof, how to actually extend a given set of LI vectors of a vector space to its basis?
Thanks.
 A: The pertinent fact here is the following:

If $\mathcal{B}$ is a (finite) linearly independent set in vector space $V$, and $w \in V \setminus \operatorname{span} \mathcal{B}$, then $\mathcal{B} \cup \{w\}$ is also linearly independent.

To prove this, suppose $\mathcal{B} = \{v_1, \ldots, v_n\}$, and suppose $a_1, \ldots, a_n, a$ are scalars such that
$$a_1 v_1 + \ldots + a_n v_n + aw = 0.$$
Note that, if $a = 0$, then linear independence of $\mathcal{B}$ implies $a_1 = \ldots = a_n = 0$ as well, and we are done. Otherwise, if $a \neq 0$, then
$$w = -\frac{a_1}{a} v_1 - \frac{a_2}{a} v_2 - \ldots - \frac{a_n}{a} \in \operatorname{span} \mathcal{B},$$
against assumption, proving the result.
What this tells us is that we can extend a linearly independent set $\mathcal{B}$ to a strictly larger set whenever $\operatorname{span} \mathcal{B} \neq V$, simply by choosing some $w \in V \setminus \operatorname{span} \mathcal{B}$, and adjoining it to $\mathcal{B}$.
The procedure for extending a linearly independent set to a basis is really this simple: keep adding vectors that are not in the span (which will maintain linear independence) until you run out of vectors to add. At that point, the span of your linearly independent set is the entire space, i.e. your set is a basis.
