<s>[INST] Matrix representation of an operator relative to the given basis A linear operator A in the three-dimensional space is determined by the transformation of the coordinates of a vector $\begin{pmatrix}
                             x\\
                             y\\
z\\
                             \end{pmatrix}$
whose coordinates are given in the standard basis, this vector after transformation becomes 
$\begin{pmatrix}
x-2y\\
y+z\\
y-x\\
                             \end{pmatrix}$.
The new basis in the space is chosen as 
$\begin{pmatrix}
1\\
1\\
1\\
                             \end{pmatrix}$, 
$\begin{pmatrix}
1\\
1\\
0\\
                             \end{pmatrix}$,
$\begin{pmatrix}
1\\
0\\
0\\
                             \end{pmatrix}$.
Find the matrix representation of this operator relative to the given basis.
 [/INST] If I understood right, $T(x,y, z)=(x-2y,y+z,y-x)$ thus $$T(1, 1, 1)=(1-2, 1+1, 1-1)=(-1, 2, 0)=0(1, 1, 1)+2(1,1,0)-3(1, 0, 0)$$
and $$T(1, 1, 0)=(-1,1,0)=0(1, 1, 1)+1(1,1,0)-2(1, 0, 0)$$
and $$T(1, 0, 0)=(1,0,-1)=-1(1, 1, 1)+1(1,1,0)+1(1, 0, 0)$$
\begin{bmatrix}
    0 & 0 & -1  \\
    2 & 1 & 1  \\
-3 & -2 & 1
  \end{bmatrix}
</s>