i har tv ̊a baser f ̈or R2, B = {
1
1

,
1
2

} och C = {
 2
−1

,
−1
3

} . Best ̈am
en matris P s ̊a att f ̈or alla vektorer v ∈ R2 g ̈aller att
P [v]B = [v]C

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.

---

### 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:
\[
\begin{aligned}
2x_1 - x_2 &= 1 \\
-x_1 + 3x_2 &= 1
\]

#### 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.

i har två baser för R2, B = { [1 1], [1 2] } och C = { [2 -1], [-1 3] }. Bestäm en matris P så att för alla vektorer v ∈ R2 gäller att P [v]B = [v]C

Learn how to find the change of basis matrix P between two bases in R2, B = {[1 1], [1 2]} and C = {[2 -1], [-1 3]}. The solution includes solving systems of equations to express the vectors in terms of new bases, and constructing matrix P. Suitable for undergraduate linear algebra students.

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