Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation defined by $$T(x_1,x_2,x_3)=(2x_1-x_2+2x_3,x_1+x_2,3x_1-x_2-x_3)$$ Consider the two bases $B=\{(1,0,0),(1,1,1),(1,-1,1)\}$ and $C=\{(1,1,2),(1,2,1),(2,1,1)\}$ of $\Bbb{R}^3$
Find the matrix representation of T which is ${R_{B,C}}({3 \times 3})$ 'with respect to' B and C.
I'm not sure If I correctly translated the question. Maybe 'with respect to' should be replaced by 'in terms of'.
The answer is acquired like this,
$T(1,0,0)=(2,1,3)=3/2(1,1,2)-1/2(1,2,1)+1/2(2,1,1)$,
and the first row of R is $\{3/2,-1/2,1\}$.
I'm confused what $R$ or $R_{B,C}$ means and why the solution work like this.
(Edit) B and C are said to be ordered bases.