The explanation about the Change of Coordinates in the uploaded image can be broken down into the following points:

1. Basis and Coordinates:

• A basis $\mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n \}$ is a set of vectors that spans the vector space $V$.
• Any vector $\mathbf{x} \in V$ can be uniquely written as a linear combination of these basis vectors: $\mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + \cdots + c_n \mathbf{b}_n$ Here, $c_1, c_2, \dots, c_n$ are the coefficients (weights) of the linear combination.

2. Coordinate Vector Representation:

• The $\mathcal{B}$-coordinates of the vector $\mathbf{x}$ are the coefficients $c_1, c_2, \dots, c_n$, arranged in a column vector: $[\mathbf{x}]_\mathcal{B} = \begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{bmatrix}$
• This vector provides the representation of $\mathbf{x}$ in the $\mathcal{B}$-basis.

3. Reconstruction of $\mathbf{x}$:

• To return to the original vector $\mathbf{x}$ in the standard basis, we use the formula: $\mathbf{x} = P_\mathcal{B} [\mathbf{x}]_\mathcal{B}$ where $P_\mathcal{B} = [\mathbf{b}_1 \; \mathbf{b}_2 \; \cdots \; \mathbf{b}_n]$ is the change-of-coordinates matrix from the basis $\mathcal{B}$ to the standard basis. This matrix is constructed by placing the basis vectors as columns.

4. Change of Basis Between Two Bases:

• If we need to change from one basis $\mathcal{B} = \{\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n\}$ to another basis $\mathcal{C} = \{\mathbf{c}_1, \mathbf{c}_2, \dots, \mathbf{c}_n\}$, the change-of-coordinates matrix from $\mathcal{C}$ to $\mathcal{B}$ is given by: [ P_{\mathcal{C} \leftarrow \mathcal{B}} = \begin{bmatrix} [\mathbf{b}1]\mathcal{C} & [\mathbf{b}2]\mathcal{C} & \cdots & [\mathbf{b}n]\mathcal{C} \end{bmatrix} ] Each column of this matrix is the coordinate vector of $\mathbf{b}_i$ in the $\mathcal{C}$-basis.

5. Key Concept:

• A change-of-coordinates matrix enables us to switch between bases and understand how a vector's representation changes under different bases.

Would you like more examples, details about applying these transformations, or a specific computation?

Related Questions:

1. How do you compute the matrix $P_\mathcal{B}$ for a given basis $\mathcal{B}$?
2. What is the significance of the invertibility of $P_\mathcal{B}$?
3. How can you find $[\mathbf{b}_i]_\mathcal{C}$ for constructing $P_{\mathcal{C} \leftarrow \mathcal{B}}$?
4. What is the geometric interpretation of a change of basis?
5. How does this relate to eigenvectors and diagonalization?

Tip:

When dealing with basis transformations, always verify the linear independence of the basis vectors to ensure $P_\mathcal{B}$ is invertible!