Being $V$ a linear space of $\Bbb K$. Show that if $v_1, v_2, v_3$ are vectors of $V$ and $𝛼_1, 𝛼_2, 𝛼_3 ∈ 𝕂$ then the set $${𝛼_2v_3 − 𝛼_3v_2, 𝛼_1v_2 − 𝛼_2v_1, 𝛼_3v_1 − 𝛼_1v_3}$$ is linearly dependent.
My attempt:
If the set is linearly dependent then there are scalars $\beta1$ $\beta2$ and $\beta3$ not all zero that: $$\beta_1(𝛼_2v_3 − 𝛼_3v_2) + \beta_2(𝛼_1v_2 − 𝛼_2v_1) + \beta_3(𝛼_3v_1 − 𝛼_1v_3) = 0 $$
I started solving the equation and I decided to isolate $v_3$, so that I can write that I can express $v_3$ as a linear combination of $v_2$ and $v_1$. But for that $\beta_1\alpha_2 + \beta_3\alpha_1$ might be different that zero $\dots$
But how can I justify that?