Linear Algebra, Theory And Applications, 2012a

12.7. EXERCISES 301
(√
√ )
2
C ([0,π])? Now let V N =span
π sin (x) , ··· , 2
π sin (Nx) . For f ∈ C ([0,π]) find
a formula for the vector in V N which is closest to f with respect to the norm determined
from the above inner product. This is called the N th partial sum of the Fourier series
of f. An important problem is to determine whether and in what way this Fourier
series converges to the function f. The norm which comes from this inner product is
sometimes called the mean square norm.
14. Consider the subspace V ≡ ker (A) where
⎛
A =
⎜
⎝
1 4 −1 −1
2 1 2 3
4 9 0 1
5 6 3 4
Find an orthonormal basis for V. Hint: You might first find a basis and then use the
Gram Schmidt procedure.
15. The Gram Schmidt process starts with a basis for a subspace {v 1 , ··· ,v n } and produces
an orthonormal basis for the same subspace {u 1 , ··· ,u n } such that
⎞
⎟
⎠
span (v 1 , ··· ,v k ) = span (u 1 , ··· ,u k )
for each k. Show that in the case of R m the QR factorization does the same thing.
More specifically, if
A = ( )
v 1 ··· v n
and if
A = QR ≡ ( )
q 1 ··· q n R
then the vectors {q 1 , ··· , q n } is an orthonormal set of vectors and for each k,
span (q 1 , ··· , q k ) = span (v 1 , ··· , v k )
16. Verify the parallelogram identify for any inner product space,
|x + y| 2 + |x − y| 2 =2|x| 2 +2|y| 2 .
Why is it called the parallelogram identity?
17. Let H be an inner product space and let K ⊆ H be a nonempty convex subset. This
means that if k 1 ,k 2 ∈ K, then the line segment consisting of points of the form
tk 1 +(1− t) k 2 for t ∈ [0, 1]
is also contained in K. Suppose for each x ∈ H, there exists Px defined to be a point
of K closest to x. Show that Px is unique so that P actually is a map. Hint: Suppose
z 1 and z 2 both work as closest points. Consider the midpoint, (z 1 + z 2 ) /2 and use the
parallelogram identity of Problem 16 in an auspicious manner.
18. In the situation of Problem 17 suppose K is a closed convex subset and that H
is complete. This means every Cauchy sequence converges. Recall from calculus a
sequence {k n } is a Cauchy sequence if for every ε > 0 there exists N ε such that
whenever m, n > N ε , it follows |k m − k n |