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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:

\[
M_{B',B'}(T) = \begin{pmatrix}
6 & 9 & 1 \\
-3 & -5 & 0 \\
-1 & -1 & 0
\end{pmatrix}
\]

This matrix represents the transformation in the new basis \( B' = \{(1,1,1), (1,2,1), (1,0,0)\} \).

Would you like further details or have any questions? Here are 5 related questions you can explore:

1. How do you calculate the inverse of a matrix, and why is it necessary for change of basis?
2. What is the geometric interpretation of changing a basis in vector spaces?
3. How would the process change if we used a different linear transformation matrix?
4. Can you explain how linear transformations preserve or alter vector spaces?
5. What is the importance of diagonalizing a matrix in the context of linear transformations?

**Tip**: When changing bases in a vector space, remember that the transformation matrices depend heavily on the relation between the original and new bases.

Let B = {(1,0,0), (0,1,0), (0,0,1)} and B' = {(1,1,1), (1,2,1), (1,0,0)} be two ordered basis of R^3. Let T: R^3 → R^3 be a linear transformation whose matrix with respect to the standard basis B in domain and B in codomain is M_B,B(T) = [[1, 1, 0], [1, -1, 0], [1, 1, 1]]. Find M_B',B'(T).

Solve the problem of finding the matrix representation of a linear transformation T: R^3 → R^3 with respect to a new basis B'. This solution involves the concepts of change of basis, matrix representation, and linear transformations, suitable for students in Grades 11-12.

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