To solve this problem, let's break it into parts:

Problem Breakdown:

1. Understand the given bases:

   • $S_1 = \{ (1, 2, 0), (1, 3, 2), (0, 1, 3) \}$
   • $S_2 = \{ (1, 2, 1), (0, 1, 2), (1, 4, 6) \}$

2. Input vector: The coordinates of vector $\vec{v}$ in basis $S_1$ are $\begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}$.

3. Tasks:

   • Determine the change of basis matrix from $S_1$ to $S_2$.
   • Use the change of basis matrix to calculate the coordinates of $\vec{v}$ in $S_2$.
   • Determine the natural (standard) coordinates of $\vec{v}$ in $\mathbb{R}^3$.

Step 1: Change of Basis Matrix

To find the change of basis matrix $P$ from $S_1$ to $S_2$, we follow this procedure:

• First, express the vectors in $S_2$ in terms of the basis $S_1$.
• Solve for the transformation matrix $P$.

The relationship between the bases can be written as: [ P \cdot [\vec{v}]{S_1} = [\vec{v}]{S_2} ]

Step 2: Compute Natural Coordinates

The vector $\vec{v}$ can also be expressed in the natural (standard) basis $\mathbb{R}^3$ using the original coordinates in $S_1$ and the vectors in $S_1$: $\vec{v}_{\text{natural}} = 2 \cdot (1, 2, 0) + 0 \cdot (1, 3, 2) + 1 \cdot (0, 1, 3).$

Let me calculate these steps in detail.import numpy as np

Basis vectors for S1 and S2

S1 = np.array([[1, 2, 0], [1, 3, 2], [0, 1, 3]]).T # Basis S1 as columns S2 = np.array([[1, 2, 1], [0, 1, 2], [1, 4, 6]]).T # Basis S2 as columns

Coordinates of v in basis S1

v_S1 = np.array([2, 0, 1])

Compute the change of basis matrix P (from S1 to S2)

First, express S2 in terms of S1

P = np.linalg.solve(S1, S2)

Compute v in basis S2 using P

v_S2 = P @ v_S1

Compute v in the natural (standard) basis using S1

v_natural = S1 @ v_S1

P, v_S2, v_natural(array([[ 2., -1., 1.], [-1., 1., 0.], [ 1., 0., 2.]]), array([ 5., -2., 4.]), array([2, 5, 3]))(array([[ 2., -1., 1.], [-1., 1., 0.], [ 1., 0., 2.]]), array([ 5., -2., 4.]), array([2, 5, 3]))success### Results:

1. Change of Basis Matrix $P$ from $S_1$ to $S_2$: \begin{bmatrix} 2 & -1 & 1 \\ -1 & 1 & 0 \\ 1 & 0 & 2 \end{bmatrix}$$
2. Coordinates of $\vec{v}$ in Basis $S_2$: \begin{bmatrix} 5 \\ -2 \\ 4 \end{bmatrix}$$
3. Coordinates of $\vec{v}$ in the Natural (Standard) Basis: \begin{bmatrix} 2 \\ 5 \\ 3 \end{bmatrix}$$

Would you like a detailed explanation of any of these steps? Here are some additional questions to explore:

1. How is the change of basis matrix calculated in general?
2. What does the matrix $P$ tell us about the relationship between $S_1$ and $S_2$?
3. How can we verify the computed $\vec{v}_{S_2}$?
4. Why do we need the natural coordinates of $\vec{v}$?
5. How can this process be extended to higher dimensions?

Tip: Understanding change of basis is key to transitioning between different coordinate systems in vector spaces.