Description
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 < f, g >=
R 1
0
f(x)g(x)dx for f, g ∈ V . Prove
that 1, x, x2
are linearly independent in V but not orthogonal.
3. Given the vectors
v1 =
Ñ
0
1
1
é
, v2 =
Ñ
1
0
1
é
, v3 =
Ñ
1
1
0
é
,
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 < p, q >=
R 1
0
p(x)q(x)dx. Use
Gram-Schmidt to find an orthonormal basis for V .
