Suppose $U$ is an upper triangular matrix and nonsingular. Say it is 3 by 3. We know there is some $A$ such that $AU = I\_3$. Let $A = [a\_1,a\_2,a\_3]$ where $a\_i$ are column vectors. Let $u\_1 = (u\_{11},0,0)^T$, $u\_2 = (u\_{12},u\_{22}, 0)^T$, $u\_3 = (u\_{13}, u\_{23}, u\_{33})^T$ be column vectors of $U$. Then, 

$$ AU = I\_3 \implies a\_iu\_i = e\_i $$

where $(e\_i)$ are standard basis. and so we have the following system of equations:

$$ a\_1u\_{11} = e\_1, a\_1 u\_{12} + a\_2 u\_{22} = e\_2, a\_1 u\_{13} + a\_2 u\_{23} + a\_3 u\_{33} = e\_3 $$

I dont understand how this system of equations follows from $AU = I\_3$. Can someone explain this procedure to me? thanks