Show that the matrix $I-uu^T$ has rank $n-1$ where $u$ is a unit vector in $R^n$ I've tried a few examples and I know that $I-uu^T$ is symmetric but I'm stuck here. Any help will be appreciated!

Note that $\rm P_u := u u^\top$ is the (rank-$1$) projection matrix that projects onto the line spanned by vector $\rm u$. Hence, ${\rm I}_n - {\rm P_u}$ is the projection matrix that projects onto the $(n-1)$-dimensional orthogonal complement of the line. Since the rank of a projection matrix is equal to its trace,
$$\mbox{rank} \left( {\rm I}_n - {\rm P_u} \right) = \mbox{tr} \left( {\rm I}_n - {\rm P_u} \right) = n - 1$$
