Transformation matrix between 2 bases Given a matrix $A = \begin{bmatrix}1 & 1 & 1\\
0 & 1 & 1\\
0 & 0 & 1
\end{bmatrix}$
 and bases to a the vector space $V$: $B=(v_{1},v_{2},v_{3}),\qquad B_{1}=(v_{1},v_{1}+v_{2},v_{1}+v_{2}+v_{3})$
Is A the transformation matrix between the basis $B$ to $B_1$?
I see that $BA=B_1$. But does it mean that it is the transformation matrix?
Thank you in advance.
 A: The transformation $x \mapsto Ax$ can be interpreted in multiple ways.
Things to remember first:


*

*The mapping $x \mapsto Mx$ for some matrix $M$ is a mapping between coordinate vectors.

*Any transformation is uniquely determined by how it transforms a set of basis vectors.


So let's look at how we could interpret $x \mapsto Ax$ in $2$ different ways:
$\#1$: Let $\begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix}$, and $\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}$ be the coordinate vectors representing $v_1$, $v_2$, and $v_3$ respectively with respect to the basis $B$.
Then we can see that the transformation $x \mapsto Ax$ maps each of these three vectors to the coordinates of $v_1,\ v_1 + v_2,$ and $v_1 + v_2 + v_3$ with respect to the $\{v_1, v_2, v_3\}$ basis.  Then we see how these coordinate vectors transform:
$A = \begin{bmatrix}1 & 1 & 1\\
0 & 1 & 1\\
0 & 0 & 1
\end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}= \begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix}$ and $A = \begin{bmatrix}1 & 1 & 1\\
0 & 1 & 1\\
0 & 0 & 1
\end{bmatrix} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}= \begin{bmatrix} 1 \\ 1 \\ 0\end{bmatrix}$ and $A = \begin{bmatrix}1 & 1 & 1\\
0 & 1 & 1\\
0 & 0 & 1
\end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}= \begin{bmatrix} 1 \\ 1 \\ 1\end{bmatrix}$.
Notice that this is just the coordinate vectors of $v_1$, $v_1 + v_2$, and $v_1 + v_2 + v_3$ with respect to the basis $B$.
So under this interpretation, the mapping $x \mapsto Ax$ transforms the vector $x$ itself without changing the basis (both the domain and image of the transformation are coordinate vectors with respect to the basis $B$).
$\#2$: Let $\begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix}$, $\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix}$, and $\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}$ be the coordinate vectors representing $v_1$, $v_1 + v_2$, and $v_1 + v_2 + v_3$, respectively, with respect to the basis $B_1$.
Again, we look at how these coordinate vectors transform (though I won't write it out, again, because they map to the same coordinate vectors).
So in this case we can think of this transformation as mapping vectors with respect to the $B_1$ basis to their coordinates in the basis $B$.
Under this interpretation, your mapping is actually the opposite of what you thought -- it maps vectors from the $B_1$ basis to their coordinates w.r.t. $B$ basis.  The one which maps vectors from $B$ to $B_1$ would be the inverse of this matrix.
Hope that helps.  :)
