[SOLVED] MATH307 Individual Homework10

Instructions: Read textbook pages 51 to 54 before working on the homework problems. Show all steps to get full credits.

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 .