The problem is to find vectors $u_1$ and $u_2$ in $\mathbb{R}^2$ such that the given matrix $S$ is the transition matrix from the basis $\{v_1, v_2\}$ to the basis $\{u_1, u_2\}$.

Given:

• $v_1 = \begin{bmatrix} 3 \\ -15 \end{bmatrix}$, $v_2 = \begin{bmatrix} -13 \\ 3 \end{bmatrix}$
• $S = \begin{bmatrix} 1 & 15 \\ -1 & -14 \end{bmatrix}$

The formula to find $u_1$ and $u_2$ is: $\begin{bmatrix} u_1 & u_2 \end{bmatrix} = \begin{bmatrix} v_1 & v_2 \end{bmatrix} \cdot S$

Calculation:

1. Construct the matrix $\begin{bmatrix} v_1 & v_2 \end{bmatrix}$: [ \begin{bmatrix} v_1 & v_2 \end{bmatrix} = \begin{bmatrix} 3 & -13 \ -15 & 3 \end{bmatrix}