Consider the optimal portfolio choice problem in the presence of \(N\) risky assets with returns \(\left(\tilde{r}_{1}, \ldots, \tilde{r}_{N}\right)\) and of a risk free asset with return \(r_{f}>0\). Suppose that there are no redundant assets in the economy in the sense that the random variables \(\left(r_{f}, \tilde{r}_{1}, \ldots, \tilde{r}_{N}\right)\) are linearly independent. Show that the optimal portfolio \(w^{*} \in \mathbb{R}^{N}\) solving problem (3.2), for a strictly increasing and strictly concave utility function \(u\), is unique.