 5.3.1: Find the least squares solution of each of the following systems: (...
 5.3.2: For each of your solutions x in Exercise 1: (a) determine the proje...
 5.3.3: For each of the following systems Ax = b, find all least squares so...
 5.3.4: For each of the systems in Exercise 3, determine the projection p o...
 5.3.5: (a) Find the best least squares fit by a linear function to the dat...
 5.3.6: Find the best least squares fit to the data in Exercise 5 by a quad...
 5.3.7: Given a collection of points (x1, y1), (x2, y2), ... , (xn, yn), le...
 5.3.8: The point (x, y) is the center of mass for the collection of points...
 5.3.9: Let A be an m n matrix of rank n and let P = A(AT A) 1AT . (a) Show...
 5.3.10: Let A be an 8 5 matrix of rank 3, and let b be a nonzero vector in ...
 5.3.11: Let P = A(AT A) 1AT , where A is an m n matrix of rank n. (a) Show ...
 5.3.12: Show that if A I O AT x r = b 0 then x is a least squares solution ...
 5.3.13: Let A Rmn and let x be a solution of the least squares problem Ax =...
 5.3.14: Find the equation of the circle that gives the best least squares c...
 5.3.15: Suppose that in the search procedure described in Example 4, the se...
Solutions for Chapter 5.3: Least Squares Problems
Full solutions for Linear Algebra with Applications  9th Edition
ISBN: 9780321962218
Solutions for Chapter 5.3: Least Squares Problems
Get Full SolutionsSince 15 problems in chapter 5.3: Least Squares Problems have been answered, more than 12006 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. Chapter 5.3: Least Squares Problems includes 15 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Column space C (A) =
space of all combinations of the columns of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.