Linear Algebra, Theory And Applications, 2012a

300 INNER PRODUCT SPACES
Show this yields an inner product on P n . Hint: Most of the axioms are obvious. The
one which says (p, p) = 0 if and only if p = 0 is the only interesting one. To verify this
one, note that a nonzero polynomial of degree no more than n − 1 has at most n − 1
zeros.
11. Let C ([0, 1]) denote the vector space of continuous real valued functions defined on
[0, 1]. Let the inner product be given as
(f,g) ≡
∫ 1
0
f (x) g (x) dx
Show this is an inner product. Also let V be the subspace described in Problem 9.
Using the result of this problem, find the vector in V which is closest to x 4 .
12. A regular Sturm Liouville problem involves the differential equation, for an unknown
function of x which is denoted here by y,
(p (x) y ′ ) ′ +(λq (x)+r (x)) y =0,x∈ [a, b]
anditisassumedthatp (t) ,q(t) > 0 for any t ∈ [a, b] and also there are boundary
conditions,
C 1 y (a)+C 2 y ′ (a) = 0
C 3 y (b)+C 4 y ′ (b) = 0
where
C1 2 + C2 2 > 0, and C3 2 + C4 2 > 0.
There is an immense theory connected to these important problems. The constant, λ
is called an eigenvalue. Show that if y is a solution to the above problem corresponding
to λ = λ 1 and if z is a solution corresponding to λ = λ 2 ≠ λ 1 ,then
∫ b
a
q (x) y (x) z (x) dx =0. (12.10)
and this defines an inner product. Hint: Do something like this:
(p (x) y ′ ) ′ z +(λ 1 q (x)+r (x)) yz = 0,
(p (x) z ′ ) ′ y +(λ 2 q (x)+r (x)) zy = 0.
Now subtract and either use integration by parts or show
(p (x) y ′ ) ′ z − (p (x) z ′ ) ′ y =((p (x) y ′ ) z − (p (x) z ′ ) y) ′
and then integrate. Use the boundary conditions to show that y ′ (a) z (a)−z ′ (a) y (a) =
0andy ′ (b) z (b) − z ′ (b) y (b) =0. The formula, (12.10) is called an orthogonality relation.
It turns out there are typically infinitely many eigenvalues and it is interesting
to write given functions as an infinite series of these “eigenfunctions”.
13. Consider the continuous functions defined on [0,π] ,C([0,π]) . Show
(f,g) ≡
∫ π
0
fgdx
{√ } ∞
2
is an inner product on this vector space. Show the functions
π sin (nx) are
n=1
an orthonormal set. What does this mean about the dimension of the vector space