Exercise 5

Example 1.4 suggests that projection breaks $\vec{v}$ into two parts, $\text{proj}_{[\vec{s}\,]}({\vec{v}\,})$ and $\vec{v}-\text{proj}_{[\vec{s}\,]}({\vec{v}\,})$, that are non-interacting. Recall that the two are orthogonal. Show that any two nonzero orthogonal vectors make up a linearly independent set.