 1.10.1E: The container of a breakfast cereal usually lists the number of cal...
 1.10.2E: In a certain region, about 6% of a city’s population moves to the s...
 1.10.3E: [M] Budget® Rent A Car in Wichita, Kansas has a fleet of about 500 ...
 1.10.4E: Clock’s moving hands At what rate is the angle between a clock’s mi...
 1.10.5E: Oil spill An explosion at an oil rig located in gulf waters causes ...
 1.10.6E: Find the first four terms of the Taylor series for the functions in...
 1.10.7E: In Exercises 5–8, write a matrix equation that determines the loop ...
 1.10.8E: In Exercise, use series to estimate the integrals’ values with an e...
 1.10.9E: In Exercise, use series to estimate the integrals’ values with an e...
 1.10.10E: In Exercise, use series to estimate the integrals’ values with an e...
 1.10.11E: In 2012 the population of California was 38,041,430, and the popula...
 1.10.12E: In Exercise, use series to estimate the integrals’ values with an e...
 1.10.13E: [M] Let M and x0 be as in Example 3.a. Compute the population vecto...
 1.10.14E: The complete elliptic integral of the first kind is the integral ,w...
Solutions for Chapter 1.10: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 1.10
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 1.10 includes 14 full stepbystep solutions. Since 14 problems in chapter 1.10 have been answered, more than 40494 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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 .

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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.