How do I prove the fact that any vector in $n$ dimensions can be written as a linear combination of $n$ separate vectors?

Question

Like $v = av_1 + bv_2 + cv_3$, where $v$ is your vector and $v_1$, $v_2$, $v_3$ are your basis.

I'm not sure where the property has come from. Googling it gives me proof about the uniqueness of such a representation but I can't find a proof on why they can even be written that way.

Answer 1

Becuse, by definition of basis, a basis $B=\{v_1,\ldots,v_n\}$ of a vector space $V$ spans $V$, which means that any $v\in V$ can be written as $\alpha_1v_1+\cdots+\alpha_nv_n$.