Math 304
Quiz 14
Summer 2006
Linear Algebra
Both problems concern the inner-product space C[0, 1] of continuous functions on the interval [0, 1] with the following inner product:
Z 1
hf, gi =
f (x)g(x) dx.
0
1. For which positive number a does the function f (x) = ax4 have norm
equal to 1?
Solution. Since kf k2 = hf, f i =
2
want a /9 = 1, so a = 3.
R1
0
h 9 i1
a2 x8 dx = a2 x9
= a2 /9, we
0
2. Find the cosine of the angle between the functions f (x) =
g(x) = 1.
√
x and
Solution. Since hf, gi = kf k kgk cos(θ), compute:
1
2
x3/2
= ,
x dx =
hf, gi =
3/2 0 3
0
2 1
Z 1
1
1
x
= ,
kf k2 =
so kf k = √ ,
x dx =
2 0 2
2
0
Z 1
kgk2 =
12 dx = 1,
so kgk = 1.
Z
1
√
0
Therefore
√
hf, gi
2/3
2 2
cos(θ) =
= √ =
.
kf k kgk 1/ 2
3
√
[The question did not ask for the value of θ, which is arccos(2 2/3).
Your calculator will give you an approximate value of either 0.34 radians
or 19.5 degrees.]
June 22, 2006
Dr. Boas