$\{x_1,x_2\}$ linearly independent. $\{x_1,x_2,u,v\}$, $\{x_1,x_2,w,z\}$ are basis => $\{u,v,w,z\}$ not a basis?

Question

Let $V$ be a vector space of $\dim(V)=4$, and $\{x_1,x_2\}$ are linearly independent in $V$.

We can complete it to a basis of $V$: $B_1=\{x_1,x_2,u,v\}$ and another one $B_1=\{x_1,x_2,w,z\}$.

Is $C=\{u,v,w,z\}$ necessarily not a basis for $V$?

Answer 1

Consider the vector space $\mathbb{R}^4$ and let $e_1, e_2, e_3, e_4$ denote the standard basis. Consider the bases $$\{e_1, e_2, e_3+e_1,e_4+e_2\}$$ and $$\{e_1, e_2, e_3+2e_1, e_4+2e_2\}.$$ Then $$\{e_3+e_1,e_4+e_2,e_3+2e_1,e_4+2e_2\}$$ is also a basis since: $$e_1 = (e_3+2e_1)-(e_3+e_1),$$ $$e_2 = (e_4+2e_2)-(e_4+e_2),$$ $$e_3 = 2(e_3+e_1)-(e_3+2e_1),$$ $$e_4 = 2(e_4+e_2)-(e_4+2e_2).$$