To find the correct transformation matrix \( A \) that relates vectors \( \mathbf{u} \) and \( \mathbf{v} \), we need to express vector \( \mathbf{v} \) as a transformation of vector \( \mathbf{u} \) using the matrix \( A \). The relationship is given as:

\( A \mathbf{u} = \mathbf{v} \).

From the diagram:

• \( \mathbf{u} = \begin{bmatrix} 4 \\ 3 \end{bmatrix} \)
• \( \mathbf{v} = \begin{bmatrix} 5 \\ 0 \end{bmatrix} \)

We need to find \( A \) such that:

\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 4 \\ 3 \end{bmatrix} = \begin{bmatrix} 5 \\ 0 \end{bmatrix} \]

This yields the following equations:

1. \( 4a + 3b = 5 \)
2. \( 4c + 3d = 0 \)

These are two linear equations in terms of the elements of matrix \( A \).

Solving these equations gives:

\[ \text{From Equation 1:} \quad a = \frac{4}{5}, \quad b = \frac{3}{5}. \] \[ \text{From Equation 2:} \quad c = -\frac{3}{5}, \quad d = \frac{4}{5}. \]

Thus, the matrix \( A \) is:

\( \begin{bmatrix} \frac{4}{5} & \frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \end{bmatrix} \)

Checking the options given, this corresponds to the first option provided in the question.

Therefore, the correct choice of matrix \( A \) is: