We are tasked with finding the matrix representation $M_{B',B'}(T)$ of a linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ with respect to the new ordered basis $B'$, given the matrix of $T$ with respect to the standard basis $B = \{(1,0,0), (0,1,0), (0,0,1)\}$.

Given:

• Standard basis $B = \{(1,0,0), (0,1,0), (0,0,1)\}$
• New basis $B' = \{(1,1,1), (1,2,1), (1,0,0)\}$
• Matrix $M_{B,B}(T) = \begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 1 & 1 & 1 \end{pmatrix}$

Strategy to find $M_{B',B'}(T)$:

1. Change of basis matrix: Find the matrix $P$ that converts vectors from the $B'$-basis to the standard basis $B$. This matrix $P$ is constructed by expressing each vector in $B'$ in terms of the standard basis.
2. Inverse change of basis matrix: Find the inverse $P^{-1}$, which converts vectors from the standard basis $B$ to the $B'$-basis.
3. Transformation to new basis: Use the formula: $M_{B',B'}(T) = P^{-1} M_{B,B}(T) P$

Let's start by calculating the change of basis matrix $P$, the inverse of $P$, and finally compute the transformed matrix. I'll begin with these calculations.The matrix representation of the linear transformation $T$ with respect to the new ordered basis $B'$ is: