288 CHAPTER 11. INNER PRODUCT SPACES

20. Using Problem 19 and Problems 17 - 18 show the projection map, P onto a closedconvex subset is Lipschitz continuous with Lipschitz constant 1. That is

|Px− Py| ≤ |x− y|

21. Give an example of two vectors in R4 or R3 x,y and a subspace V such that x · y = 0but Px·Py ̸= 0 where P denotes the projection map which sends x to its closest pointon V .

22. Suppose you are given the data, (1, 2) , (2, 4) , (3, 8) , (0, 0) . Find the linear regressionline using the formulas derived above. Then graph the given data along with yourregression line.

23. Generalize the least squares procedure to the situation in which data is given and youdesire to fit it with an expression of the form y = af (x)+bg (x)+c where the problemwould be to find a, b and c in order to minimize the error. Could this be generalizedto higher dimensions? How about more functions?

24. Let A ∈ L (X,Y ) where X and Y are finite dimensional vector spaces with the dimen-sion 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 {xi}ri=1 be a basis for ker (A)

and let {xi}ri=1 ∪ {yi}n−ri=1 be a basis for X. Then show that {Ayi}n−r

i=1 is linearlyindependent 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 thefollowing 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, {Axj}rj=1 of A (Fn) and verify {A∗Axj}rj=1

is a basis for A∗A (Fn) .

26. Let A be a real m × n matrix and let A = QR be the QR factorization with Qorthogonal and R upper triangular. Show that there exists a solution x to the equation

RTRx = RTQTb

and that this solution is also a least squares solution defined above such that ATAx =ATb.

11.8 The Determinant and Volume

The determinant is the essential algebraic tool which provides a way to give a unified treat-ment of the concept of p dimensional volume of a parallelepiped in RM . Here is the definitionof what is meant by such a thing.

Definition 11.8.1 Let u1, · · · ,up be vectors in RM ,M ≥ p. The parallelepiped determinedby these vectors will be denoted by P (u1, · · · ,up) and it is defined as

P (u1, · · · ,up) ≡

p∑

j=1

sjuj : sj ∈ [0, 1]

 = UQ, Q = [0, 1]p