To find the transition matrix $P_{C \to B}$ from the basis $C = \{ \vec{v}_1, \vec{v}_2 \}$ to the basis $B = \{ \vec{u}_1, \vec{u}_2 \}$, we need to express each vector in $C$ in terms of the basis vectors in $B$.

Here are the steps:

1. Express $\vec{v}_1$ in terms of $\vec{u}_1$ and $\vec{u}_2$: Find scalars $a$ and $b$ such that: $\vec{v}_1 = a\vec{u}_1 + b\vec{u}_2$ This can be written as a system of linear equations.

2. Express $\vec{v}_2$ in terms of $\vec{u}_1$ and $\vec{u}_2$: Find scalars $c$ and $d$ such that: $\vec{v}_2 = c\vec{u}_1 + d\vec{u}_2$ Similarly, this will lead to another system of linear equations.

3. Form the matrix: The matrix $P_{C \to B}$ will have the form: $P_{C \to B} = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ where $a, b$ are the coefficients for $\vec{v}_1$ and $c, d$ are the coefficients for $\vec{v}_2$.

Let's solve for the coefficients by setting up the systems of equations and solving them.The transition matrix $P_{C \to B}$ from the basis $C = \{ \vec{v}_1, \vec{v}_2 \}$ to the basis $B = \{ \vec{u}_1, \vec{u}_2 \}$ is:

$P_{C \to B} = \begin{bmatrix} -\frac{1}{3} & 1 \\ -\frac{1}{3} & 2 \end{bmatrix}$

This matrix represents how vectors in the basis $C$ can be expressed in terms of the basis $B$.

Would you like more details on the process or have any other questions?

Here are 5 related questions you might find helpful:

1. How do you find the transition matrix from one basis to another in higher dimensions?
2. What is the geometric interpretation of a transition matrix?
3. How can you verify that a given matrix is indeed a valid transition matrix?
4. What is the inverse of a transition matrix and how is it related to the original bases?
5. How can transition matrices be applied in computer graphics transformations?

Tip: The determinant of a transition matrix indicates how the area (or volume in higher dimensions) is scaled when transitioning between bases.