To solve this, we need to follow these steps:

1. Understand the Problem

We are given a linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^2$, which means the matrix of the transformation goes from 3-dimensional space to 2-dimensional space.

• Old basis of $\mathbb{R}^3$: $\{(0,1,1), (1,0,1), (1,1,0)\}$
• Old basis of $\mathbb{R}^2$: $\{(1,1), (0,1)\}$
• We are given the matrix of $T$ relative to these bases, but it hasn't been explicitly provided yet.

Now, we are asked to find the matrix of $T$ relative to new bases:

• New basis of $\mathbb{R}^3$: $\{(0,1,1), (1,0,1), (0,0,1)\}$
• New basis of $\mathbb{R}^2$: $\{(1,1), (0,1)\}$ (same as before)

2. Strategy to Solve the Problem

Step 1: Change of Basis for $\mathbb{R}^3$

To convert the matrix from the old basis of $\mathbb{R}^3$ to the new basis, we need the change of basis matrix from the old basis to the new basis in $\mathbb{R}^3$.

Let’s denote the old basis of $\mathbb{R}^3$ as: $B_{\text{old}} = \{v_1, v_2, v_3\} = \{(0,1,1), (1,0,1), (1,1,0)\}$ Let the new basis of $\mathbb{R}^3$ be: $B_{\text{new}} = \{w_1, w_2, w_3\} = \{(0,1,1), (1,0,1), (0,0,1)\}$

To find the change of basis matrix, we need to express each vector in the new basis $B_{\text{new}}$ as a linear combination of the vectors in the old basis $B_{\text{old}}$.

Step 2: Compute the Change of Basis Matrix for $\mathbb{R}^3$

The change of basis matrix from $B_{\text{old}}$ to $B_{\text{new}}$ is the matrix $P_{\text{old} \to \text{new}}$ such that: [ P_{\text{old} \to \text{new}} = [ [w_1]{B{\text{old}}}, [w_2]{B{\text{old}}}, [w_3]{B{\text{old}}}] ]

Step 3: Find the Matrix of $T$ in New Basis

Once we have the change of basis matrix $P_{\text{old} \to \text{new}}$, we can compute the matrix of $T$ relative to the new basis by the following formula: [ [T]{B{\text{new}}} = P_{\text{old} \to \text{new}}^{-1} [T]{B{\text{old}}} ]

3. Implementation

We need:

1. The explicit matrix of $T$ relative to the old basis.
2. The change of basis matrix $P_{\text{old} \to \text{new}}$.
3. The inverse of $P_{\text{old} \to \text{new}}$.

If you provide the matrix of $T$ relative to the old basis, I can help you compute the matrix relative to the new basis step by step.

Would you like to proceed with more details on the calculations, or do you have any specific questions?

Here are 5 related questions for better understanding:

1. How do you calculate the change of basis matrix between two bases?
2. Why is the change of basis matrix important for linear transformations?
3. How do you invert a 3x3 matrix, and why is it necessary here?
4. What is the geometric interpretation of changing bases for a linear transformation?
5. How can you verify that the new transformation matrix is correct?

Tip: In any transformation problem involving a change of basis, always ensure to use the correct order for the basis vectors in your calculations.