1. Problem 6 on page 56.
2. Let V be the vector space of all real coefficient polynomials over the interval
[0,1], define an inner product . Prove that 1,x,x2 are linearly independent in V but not orthogonal.
3. Given the vectors
Ñ0é Ñ1é Ñ1é
v1 = 1 ,v2 = 0 ,v3 = 1 ,
1 1 0
find the projection of v1,v2 along v3 respectively, and then use them to find the projection of 2v1 + v2 along v3.
4. Let V be the vector space of all real coefficient polynomials over [0,1] with degree no more than 1. One can prove that 1,x over [0,1] form a basis of
V . Let p,q ∈ V , define an inner product . Use
Gram-Schmidt to find an orthonormal basis for V .
