Math 304 (Spring 2015) - Homework 8
Problem 1.
Determine whether the following sets of vectors form an orthonormal basis
of R2 .
(a) {(1, 0)T , (0, 1)T }
√ √ T
T
3 1
3
−1
(b)
,
, 2, 2
2 2
cos θ
− sin θ
(c)
,
sin θ
cos θ
Solution:
(a) Yes.
(b) Yes.
(c) Yes.
Problem 2.
Let {u1 , u2 , u3 } be an orthonormal basis for an inner product space V and
let
w = u1 + 2u2 + 2u3 and v = u1 + 7u3
Determine the value of each of the following:
(a) hw, vi
(b) kwk and kvk
(c) the angel between w and v.
Solution:
(a) hw, vi = 15
(b) kwk = 3 and kvk =
√
√
50 = 5 2.
(c)
hw, vi
1
=√
kwkkvk
2
So the angle between w and v is π/4.
cos θ =
1
Problem 3.
Given the basis {(1, 2, −2)T , (4, 3, 2)T , (1, 2, 1)T } for R3 , use the GramSchmidt process to obtain an orthonormal basis.
Solution: Let us denote
v1 = (1, 2, −2)T , v2 = (4, 3, 2)T , v3 = (1, 2, 1)T .
u1 =
v1
kv1 k
= ( 31 , 23 , − 32 )T .
Compute v2 − hv2 , u1 iu1 =
10 5 10 T
, ,
,
3 3 3
rescale to make it length one:
u2 = ( 32 , 13 , 23 )T
Compute v3 − hv3 , u1 iu1 − hv3 , u1 iu1 , then rescale to make it length
one. We get
u3 = (− 23 , 23 , 13 )T
Problem 4.
Let

3 −1
2
A = 4
0
2
 
0

and v = 20 .
10

(a) Find an orthonormal basis of the column space of A.
(b) Find the projection of v onto the column space of A.
Solution:
(a) An orthonormal basis of A is
u1 =
T
3 4
, , 0 , u2
5 5
T
= − 5√4 2 , 5√3 2 , √12
(b) The projection of v onto the column space of A is
hv, u1 iu1 + hv, u2 iu2 = ( 54 , 97
, 11)T
5
2
Problem 5.
(Legendre Polynomials) Let P2 = {all polynomials of degree ≤ 2}.
We define the following inner product on P2 :
Z 1
p(x)q(x)dx.
hp, qi =
−1
Start with a basis {1, x, x2 } of P2 , use the Gram-Schmidt process to obtain
an orthonormal basis.
Solution:
h1, 1i = 2
so u1 =
√1 .
2
x − hx, u1 iu1 = x − 0 = x
and
hx, xi =
so u2 =
q
3
2
q
2
.
3
x.
x2 − hx2 , u1 iu1 − hx2 , u2 iu2 = x2 −
so u3 =
q
45
(x2
8
− 13 ).
3
1
3