Let V be a finite dimensional vector space over a field F (dimension of V is necessarily > 3). Let $v_{1} , v_{2},v_{3},w_{1} , w_{2},w_{3} \in V$ such that $\{ v_{1} , v_{2},v_{3} \}$ is a linearly independent set, $\{ w_{1} , w_{2},w_{3} \}$ is a linearly independent set, and also $\{ v_{1} , w_{1} \}$, $\{ v_{2},w_{2} \}$, $\{ v_{3} ,w_{3} \}$ are linearly independent sets.
Then if there exists $w_{4} \in V$ such that $\{ w_{1} , w_{2},w_{3}, w_{4} \}$ is a linearly independent set then whether the set $\{ v_{1} , v_{2},v_{3}, w_{4} \}$ is also linearly independent or not?
My attempt:
I could not come up with anything explicitly, my intuitive idea says the answer might be yes (but that intuition relies on actually $\Bbb R^n$) so for arbitrary vector space , couldn't really come up with anything.
Thanks in advance for helping to solve the problem.
The answer is no.
Take the standard basis $e_1, e_2, e_3, e_4$ for $\mathbb{R}^4$ and let $v_i = e_{i+1}$ and $w_i = e_i$. So the conditions are fulfilled, but $\{v_1, v_2, v_3, w_4\}$ is not a linearly independent set.