Linear Algebra, Theory And Applications, 2012a

302 INNER PRODUCT SPACES
This is called a minimizing sequence. Show there exists a unique k ∈ K such that
lim n→∞ |k n − k| and that k = Px. That is, there exists a well defined projection map
onto the convex subset of H. Hint: Use the parallelogram identity in an auspicious
manner to show {k n } is a Cauchy sequence which must therefore converge. Since K
is closed it follows this will converge to something in K which is the desired vector.
19. Let H be an inner product space which is also complete and let P denote the projection
map onto a convex closed subset, K. Show this projection map is characterized by
the inequality
Re (k − Px,x− Px) ≤ 0
for all k ∈ K. That is, a point z ∈ K equals Px if and only if the above variational
inequality holds. This is what that inequality is called. This is because k is allowed
to vary and the inequality continues to hold for all k ∈ K.
20. Using Problem 19 and Problems 17 - 18 show the projection map, P ontoaclosed
convex subset is Lipschitz continuous with Lipschitz constant 1. That is
|Px− Py|≤|x − y|
21. Give an example of two vectors in R 4 x, y and a subspace V such that x · y = 0 but
P x·P y ≠0whereP denotes the projection map which sends x to its closest point on
V .
22. Suppose you are given the data, (1, 2) , (2, 4) , (3, 8) , (0, 0) . Find the linear regression
line using the formulas derived above. Then graph the given data along with your
regression line.
23. Generalize the least squares procedure to the situation in which data is given and you
desire to fit it with an expression of the form y = af (x)+bg (x)+c where the problem
would be to find a, b and c in order to minimize the error. Could this be generalized
to higher dimensions? How about more functions?
24. Let A ∈L(X, Y )whereX and Y are finite dimensional vector spaces with the dimension
of X equal to n. Define rank (A) ≡ dim (A (X)) and nullity(A) ≡ dim (ker (A)) .
Show that nullity(A)+rank(A) =dim(X) . Hint: Let {x i } r i=1
be a basis for ker (A)
and let {x i } r i=1 ∪{y i} n−r
i=1 be a basis for X. Then show that {Ay i} n−r
i=1
is linearly
independent and spans AX.
25. Let A be an m×n matrix. Show the column rank of A equals the column rank of A ∗ A.
Next verify column rank of A ∗ A is no larger than column rank of A ∗ . Next justify the
following inequality to conclude the column rank of A equals the column rank of A ∗ .
rank (A) =rank (A ∗ A) ≤ rank (A ∗ ) ≤
=rank (AA ∗ ) ≤ rank (A) .
Hint: Start with an orthonormal basis, {Ax j } r j=1 of A (Fn ) and verify {A ∗ Ax j } r j=1
is a basis for A ∗ A (F n ) .
26. Let A be a real m × n matrix and let A = QR be the QR factorization with Q
orthogonal and R upper triangular. Show that there exists a solution x to the equation
R T Rx = R T Q T b
and that this solution is also a least squares solution defined above such that A T Ax =
A T b.