In this problem, you are asked to find the change of basis matrices between two different bases $B$ and $C$ for $\mathbb{R}^2$. Let's go through why parts (a) and (b) yield the same matrix in this case.

Understanding Change of Basis Matrix

The change of basis matrix from a basis $B$ to a basis $C$, denoted $[\text{id}]_B^C$, represents how we can express each vector in basis $B$ in terms of basis $C$. Similarly, $[\text{id}]_C^B$ would be the matrix to express each vector in basis $C$ in terms of basis $B$.

To calculate $[\text{id}]_B^C$ (from $B$ to $C$):

1. Take each vector in $B$ and express it as a linear combination of vectors in $C$.
2. The coefficients of these combinations form the columns of $[\text{id}]_B^C$.

To calculate $[\text{id}]_C^B$ (from $C$ to $B$):

1. Take each vector in $C$ and express it as a linear combination of vectors in $B$.
2. The coefficients of these combinations form the columns of $[\text{id}]_C^B$.

Why They Are the Same in This Case

Since the change of basis matrix $[\text{id}]_B^C$ is typically the inverse of $[\text{id}]_C^B$, we would expect different matrices for parts (a) and (b). However, if the matrix $[\text{id}]_B^C$ is orthogonal (meaning it represents a rotation or reflection in $\mathbb{R}^2$), its inverse will be the same as its transpose, and in this specific case, it may coincide with $[\text{id}]_C^B$.

In your problem, the given entries of the matrix $[\text{id}]_B^C$ and $[\text{id}]_C^B$ suggest that they are the same because:

1. The vectors in $B$ and $C$ are related in a way that causes the transition matrices to be symmetric.
2. The bases might be aligned such that expressing one basis in terms of the other yields the same values in both directions.

Without calculating explicitly, if the relationship between $B$ and $C$ is symmetric or involves an orthogonal transformation, then both matrices can indeed be the same.

Would you like a step-by-step calculation of these matrices, or do you have specific questions about the concept?

Related Questions to Expand Understanding:

1. How do we calculate the change of basis matrix if the bases are not orthogonal?
2. What are the properties of orthogonal matrices and their inverses?
3. Can a change of basis matrix ever be equal to its inverse in other cases?
4. How would the process differ if we were working in $\mathbb{R}^3$ instead of $\mathbb{R}^2$?
5. How can you verify if two bases are related by an orthogonal transformation?

Tip:

When working with change of basis matrices, always consider the relationship between the bases—orthogonal transformations often yield symmetric properties.