91Ó°ÊÓ

Let \(\mathrm{u}=(4,0,1,2,0), \mathrm{v}=(2,1,-1,1,1) .\) Find: a) \(\mathrm{u} \cdot \mathrm{v} ; \mathrm{b})\|\mathrm{u}\|,\|\mathrm{v}\| ; \mathrm{c}\) ) the projection of \(\mathrm{u}\) onto \(\mathrm{v}\) and the projection of u orthogonal to \(\mathrm{v}\).

Show that the functions \(1, \cos \pi \mathrm{x}, \cos 2 \pi \mathrm{x}, \ldots, \cos \mathrm{n} \pi \mathrm{x}\), form an orthogonal set over \([0,1]\). Then normalize them to obtain an orthogonal set.