How to complete a span

Question

Hey I have a simple question.

Say I have vector space V (dimV > 1). B = Span{v1,v2} and C = span{u1}. I have to find a vector u2 such that span{u1,u2} = span{v1,v2} = B.

How do I do it ? thanks

Answer 1

This is not always possible. For instance, take $V = \mathbb{R}^3$ and let $e_1, e_2, e_3$ be the standard basis for $\mathbb{R}^3$. Now say I take $B = \text{span}(e_1, e_2)$ and $C = \text{span}(e_3)$. Then $C$ contains $e_3$ while $B$ does not. Hence there is no vector $u_2$ such that $\text{span}(e_3, u_2) = B$.

If you know that $u_1$ is in $B$, however, and $u_1$ is nonzero, then either $\text{span}(u_1) = B$ already, or any $u_2 \in B \setminus \text{span}(u_1)$ will satisfy $\text{span}(u_1, u_2) = \text{span}(v_1, v_2)$.