Given a vector w, the inner product of R^n is defined by:
<x,y>=Summation from i=1 to n (xi,yi,wi)
[a] Using this equation with weight vector w=(1/4,1/2,1/4)^t to define an inner product for R^3 and let x=(1,1,1)^T and y=(-5,1,3)^T
Show that x and y are orthogonal with respect to this inner product. Compute the values of ||x|| and ||y|| with respect to this inner product.
[b]In C[0,1], with inner product defined above, consider the vectors 1 and x. Find the angle theta between 1 and x. Determine the vector projection p of 1 onto x and verify that 1-p is orthogonal to p.
Inner products are calculated and vector relations proven.
The solution is detailed and well presented.