OK\\ Exercise 7\\ Put E = RIX] 1 - Show that | P ( t ) ( t ) at is a well defined inner product . Work 2 - Find an orthonormal basis for this inner product in F = 1R2 [* ] 3 - Calculate the orthogonal projection of *3 on F . DO Exercise 8 B = \\ On E = COO ( 1 0 , IT ] ; RR ) , we define the inner product } } ( t ) 9 ( 1 ) at . Put F - NE E I F"' + 8 - 0). ~ 1 - Show that F is a vector space\\ 1 : 0 - 8^ _ OF ( 21 + 9 ) = ] FIFI + F ( 9) 2 - Find an orthonormal basis of F Projection matrix` 3 - Given I ( t ) = ton [O. IT ) , find the orthogonal projection of I on !` Exercise 9 Given an Euclidean space E , and a projection p on some subspace F of E. Show that :" P is an orthogonal projection VI EE.| 7 ( 20 ) | |