Let $\{ \vec{v}_1, \ldots, \vec{v}_n \}$ form an orthonormal basis for $\mathbb{R}^n$ and let $$ \vec{X} = \begin{pmatrix} X_1 \\ \vdots \\ X_n \end{pmatrix} $$ be a vector where each $X_i \sim \mathcal{N}(0,1)$. Let $Y_i = \vec{X}^t v_i$ for $1 \leq i \leq n$. How could we show that the $Y_i$'s are independent of each other?
Is it necessary to create a joint density function of each $Y_i,Y_j$ pair or is there an easier way since each $Y_i$ is simply a linear combination of $n$ random variables with standard normal distributions?