Consider the vector space $\mathbb{R^3}$ and the bases $U=\{e\_1,e\_2,e\_3\}$ and $V=\{e\_1, e\_1+e\_2, e\_1+e\_2+e\_3\}$. I want to find the basis transition matrix $A$ from $U$ into $V$ satisfying $V=AU$. Also i want to find the coordinate vectors of the given vector $(1,2,3) \in \mathbb{R^3}$ with respect to the bases $U$ and $V$, respectively, and relate these vectors with the matrix $A$.

What I have done: I define a map $T:U \longrightarrow V$ such that $T(e\_1)=e\_1$, $T(e\_2)=e\_1 + e\_2$ and $T(e\_3)=e\_1 + e\_2 + e\_3$. I've denoted $e\_1=v\_1$, $e\_1+e\_2=v\_2$ and $e\_1 + e\_2 + e\_3 =v\_3$. Then, I've obtained that $e\_1=v\_1$, $e\_2=v\_2-v\_1$ and $e\_3=v\_3-v\_2$. I guess that the transition matrix A will be that 
$
 A=
 \left[ {\begin{array}{cc}
 1 & 1 & 1\\
 0 & 1 & 1\\
 0 & 0 & 1\\
 \end{array} } \right]
$. Here I am confused. I suppose, something is wrong. Is there any algorithm to solve this type questions?