how I do I find the matrices of these linear transformattions I asked this questions two days ago on Stackoverflow, and was advised to migrate it here.
The problem is the way the bases are given is not in the usual way. It is rather confusing. Help will be appreciated.
Going through the text on Linear Algebra by A. O. Morris (2nd edition) I am trying to understand something on linear transformations. There is a problem where the R-bases of $U$ and $V$ are given as
$\{u_1, u_2\}$ and $\{v_1,v_2,v_3\}$ respectively and the linear transformation from U to V is given by
$$
    T(u_1)=v_1+2v_2-v_3
$$
$$
    T(u_2)=v_1-v_2
$$
The problem is to:
a) find the matrix of $T$ relative to these bases,
b) the matrix relative to the R-bases
$\{-u_1+u_2,2u_1-u_2\}$ and     $\{v_1,v_1+v_2,v_1+v_2+v_3\}$
and
c) the relationship between the two matrices.
From the very good treatment of the subject here Determine the matrix relative to a given basis I figured out that the first matrix $T$ has columns
\begin{bmatrix}
     1 & 1\\
    \ 2 & -1\\
    -1 &  0
\end{bmatrix}
then for part b) I figure out Matrix $A$ which I take to be the transform of ordered basis of $U$ to standard basis of $U$ as
\begin{bmatrix}
    -1 & 2\\
    1 & -1
\end{bmatrix}
Matrix $B$ which I take to be the transform of ordered basis of $V$ to standard basis of $V$ as
\begin{bmatrix}
    1 &1 &1\\
    0 & 1& 1\\
    0  & 0 & 1
\end{bmatrix}
I then find inverse of $B$
\begin{bmatrix}
    0 & -1 & 1\\
    0 & 1 & 1\\
    0 & 0 & 1
\end{bmatrix}
and form the product
$B^{-1}CA$ as the answer to part c).
\begin{bmatrix}
    4 &-7\\
    -2 & 3\\
    1 & -2
\end{bmatrix}
Somehow I do not seem to get it. I am completely at sea. I would appreciate help to understand this. My sincere apologies for not being able to use Latex.

Your work seems correct, but you have done a mistake in the inverse of $B$.
$$
B^{-1}=
\begin{bmatrix}
1 & -1 &0\\
0 & 1 &-1\\
0 & 0 & 1
\end{bmatrix}
$$
