To find the matrix $P$ such that for all vectors $v \in \mathbb{R}^2$, the following relation holds:

[ P [v]_B = [v]_C ]

where $[v]_B$ and $[v]_C$ are the coordinate vectors of $v$ with respect to the bases $B$ and $C$, respectively, we need to express the transformation between these two bases.

Steps:

1. Write the given bases:

   The basis $B$ is given as: $B = \left\{ \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \end{pmatrix} \right\}$

   The basis $C$ is given as: $C = \left\{ \begin{pmatrix} 2 \\ -1 \end{pmatrix}, \begin{pmatrix} -1 \\ 3 \end{pmatrix} \right\}$

2. Find the matrix representing the change of basis from $B$ to $C$:

   We want to find the matrix $P$ such that for any vector $v \in \mathbb{R}^2$, we have: [ P [v]_B = [v]_C ]

   To find $P$, we express the vectors in $B$ in terms of the vectors in $C$.

3. Express $B$-basis vectors in terms of $C$-basis vectors:

   Let’s find how the vectors in $B$ are expressed as linear combinations of the vectors in $C$. We need to solve the system for both vectors in $B$.

   • For $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$, solve: $\begin{pmatrix} 1 \\ 1 \end{pmatrix} = x_1 \begin{pmatrix} 2 \\ -1 \end{pmatrix} + x_2 \begin{pmatrix} -1 \\ 3 \end{pmatrix}$

   • For $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$, solve: $\begin{pmatrix} 1 \\ 2 \end{pmatrix} = y_1 \begin{pmatrix} 2 \\ -1 \end{pmatrix} + y_2 \begin{pmatrix} -1 \\ 3 \end{pmatrix}$

4. Form the matrix $P$:

   Once the coefficients $x_1, x_2, y_1, y_2$ are found, the matrix $P$ will have these coefficients as columns: $P = \begin{pmatrix} x_1 & y_1 \\ x_2 & y_2 \end{pmatrix}$

Let’s solve these two systems of equations and construct the matrix.

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Solving the systems:

1. For the vector $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$:

$\begin{pmatrix} 1 \\ 1 \end{pmatrix} = x_1 \begin{pmatrix} 2 \\ -1 \end{pmatrix} + x_2 \begin{pmatrix} -1 \\ 3 \end{pmatrix}$ This gives the system:

2x_1 - x_2 &= 1 \\ -x_1 + 3x_2 &= 1$$ #### 2. For the vector $$\begin{pmatrix} 1 \\ 2 \end{pmatrix}$$: $$\begin{pmatrix} 1 \\ 2 \end{pmatrix} = y_1 \begin{pmatrix} 2 \\ -1 \end{pmatrix} + y_2 \begin{pmatrix} -1 \\ 3 \end{pmatrix}$$ This gives the system: $$\begin{aligned} 2y_1 - y_2 &= 1 \\ -y_1 + 3y_2 &= 2$$ I will now solve these two systems to find $$P$$.The matrix $$P$$ that transforms the coordinate vectors from basis $$B$$ to basis $$C$$ is: $$P = \begin{pmatrix} \frac{4}{5} & 1 \\ \frac{3}{5} & 1 \end{pmatrix}$$ This matrix satisfies the relation $$P [v]_B = [v]_C$$ for any vector $$v \in \mathbb{R}^2$$. Would you like a detailed explanation of the steps, or do you have any questions? Here are five related questions to deepen your understanding: 1. How do we interpret the columns of matrix $$P$$ geometrically? 2. How would you find the inverse matrix $$P^{-1}$$ and interpret its role? 3. What would the matrix $$P$$ look like for a different set of bases in $$\mathbb{R}^2$$? 4. How does changing the order of the bases affect the resulting matrix $$P$$? 5. Can this method be extended to higher-dimensional vector spaces like $$\mathbb{R}^3$$? **Tip:** The change of basis matrix $$P$$ is not unique to $$\mathbb{R}^2$$; it can be generalized to any finite-dimensional vector space.