I can now see the image. The problem is about finding a matrix $[T]_{\mathcal{B'}}^{\mathcal{B}}$ with respect to given bases.

The steps typically involved in such problems include:

1. Identifying the linear operator $T$ and its matrix representation $[T]_{\mathcal{B}}$ with respect to one basis.
2. Changing the basis from $\mathcal{B}$ to $\mathcal{B'}$ using a transition matrix $P$.
3. Using the formula for changing the matrix representation of a linear operator: [ [T]{\mathcal{B'}}^{\mathcal{B}} = P^{-1} [T]{\mathcal{B}} P ] where $P$ is the matrix of transition from basis $\mathcal{B}$ to $\mathcal{B'}$.

I will now work through the specific details of this problem using the given matrices.It seems there is an issue with the matrices that resulted in a singular matrix error, meaning one of the matrices involved in solving for the transition matrix $P$ is not invertible.

Let me revisit the problem. The transition matrix $P$ between two bases is generally found by expressing one basis in terms of the other. I will try an alternative approach to compute $P$ and continue solving the problem.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to proceed with any other questions or perhaps explore a different problem?

Here are some relative questions to help further:

1. How do transition matrices relate to basis changes in linear algebra?
2. Can you explain the concept of matrix inverses and their role in basis transformations?
3. What are the properties of singular matrices, and why can’t they be inverted?
4. How do we interpret linear operators in different bases?
5. What is the significance of the matrix representation of a linear operator?

Tip: Always check if a matrix is singular (non-invertible) before attempting to use it in operations that require an inverse!