 7.3.1E: In Exercises 1 and 2, find the change of variable x = Py that trans...
 7.3.2E: In Exercises 1 and 2, find the change of variable x = Py that trans...
 7.3.3E: In Exercises 3–6, find (a) the maximum value of Q(x) subject to the...
 7.3.4E: In Exercises 3–6, find (a) the maximum value of Q(x) subject to the...
 7.3.5E: In Exercises 3–6, find (a) the maximum value of Q(x) subject to the...
 7.3.6E: In Exercises 3–6, find (a) the maximum value of Q(x) subject to the...
 7.3.7E: Find a unit vector x in R3 at which Q(x) is maximized, subject to x...
 7.3.8E: Find a unit vector x in R3 at which Q(x) is maximized, subject to x...
 7.3.9E: Find the maximum value of subject to the constraint (Do not go on t...
 7.3.10E: Find the maximum value of subject to the constraint (Do not go on t...
 7.3.11E: Suppose x is a unit eigenvector of a matrix A corresponding to an e...
 7.3.12E: Let be any eigenvalue of a symmetric matrix A. Justify the statemen...
 7.3.13E: Let A be an n × n symmetric matrix, let M and m denote the maximum ...
 7.3.14E: In Exercises 14–17, follow the instructions given for Exercises 3–6...
 7.3.15E: In Exercises 14–17, follow the instructions given for Exercises 3–6...
 7.3.16E: In Exercises 14–17, follow the instructions given for Exercises 3–6.
 7.3.17E: In Exercises 14–17, follow the instructions given for Exercises 3–6.
Solutions for Chapter 7.3: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 7.3
Get Full SolutionsChapter 7.3 includes 17 full stepbystep solutions. Since 17 problems in chapter 7.3 have been answered, more than 35367 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.