Express vector $\mathbf v$ as linear combination of $\mathbf a_1$, $\mathbf a_2$, ..., $\mathbf a_n$

Question

Let $\mathbf a_1=(a_{11},a_{12},\dots,a_{1n})$, $\mathbf a_2=(a_{21},a_{22},\dots,a_{2n})$, ..., $\mathbf a_n=(a_{n1},a_{n2},\dots,a_{nn})$ be $n$ linearly independent vectors.

Given $\mathbf v = \alpha_1 \mathbf a_1+\alpha_2 \mathbf a_2+\dots+\alpha_n \mathbf a_n$, how do I find $\alpha_1,\alpha_2,\dots,\alpha_n$ explicitly in terms of $a_{11},a_{12},\dots,a_{nn}$?