Let $\mathbf{u}, \mathbf{v}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^n \backslash \{0\}$. Show that $\mathbf{u v}^T = \mathbf{x y}^T$ if and only if there exists $\lambda \in \mathbb{R} \backslash \{0\}$ such that
$$\mathbf{u} = \lambda \mathbf{x}, \quad \text{and} \quad \mathbf{v} = \frac{1}{\lambda} \mathbf{y}$$