Let $\{v_1,..,v_m\}$ and $\{w_1,..,w_n\}$ be linearly independent sets such that $w_j \notin$ span($v_1,..,v_m$), is $v_1,..,v_m,w_1,..,w_n$ lin.ind?

Question

Specifically, suppose that $v_1,...,v_m$ is a linearly independent list of vectors in some vector space $V$. Further, suppose that $w_1,...,w_n \in V$ is also linearly independent such that $w_j \notin \text{span}(v_1,...,v_m)$ for all j. Does it follow that $v_1,...,v_m,w_1,...,w_n$ is linearly independent?

Answer 1

It need not to be true. Consider $v=(1,2)$, $w_{1}=(1,0)$ and $w_{2}=(0,1)$ in $\Bbb{R^2}$.