To find the coordinates of a vector with respect to the basis $C$ using the transition matrix $P_{C \leftarrow B}$, you would apply the inverse of the transition matrix $P_{C \leftarrow B}$ to the coordinates of the vector in the basis $B$.

Given:

• $\mathbf{u}_1 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$
• $\mathbf{u}_2 = \begin{bmatrix} 0 \\ -3 \end{bmatrix}$
• $\mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \end{bmatrix}$
• $\mathbf{v}_2 = \begin{bmatrix} -3 \\ 2 \end{bmatrix}$

And the transition matrix $P_{C \leftarrow B} = \begin{bmatrix} -\frac{11}{2} & \frac{9}{2} \\ \frac{3}{2} & -\frac{3}{2} \end{bmatrix}$.

To find the coordinates of a vector $\mathbf{x}$ in the basis $C$, denoted $[\mathbf{x}]_C$, we can apply the following formula:

[ [\mathbf{x}]C = P{C \leftarrow B} \cdot [\mathbf{x}]_B ]

Now, you need to provide the vector $\mathbf{x}$ in the basis $B$ (denoted as $[\mathbf{x}]_B$) for which you want to find the coordinates in the basis $C$. Once provided, we can use the above formula to find the coordinates of the vector in the basis $C$.

Please provide the vector in the basis $B$ you wish to convert to the basis $C$.