 8.2.1E: In Exercises 1–6, determine if the set of points is affinely depend...
 8.2.2E: In Exercises 1–6, determine if the set of points is affinely depend...
 8.2.3E: In Exercises 1–6, determine if the set of points is affinely depend...
 8.2.4E: In Exercises 1–6, determine if the set of points is affinely depend...
 8.2.5E: In Exercises 1–6, determine if the set of points is affinely depend...
 8.2.6E: In Exercises 1–6, determine if the set of points is affinely depend...
 8.2.7E: In Exercises 7 and 8, find the barycentric coordinates of p with re...
 8.2.8E: In Exercises 7 and 8, find the barycentric coordinates of p with re...
 8.2.9E: In Exercises 9 and 10, mark each statement True or False. Justify e...
 8.2.10E: In Exercises 9 and 10, mark each statement True or False. Justify e...
 8.2.11E: Explain why any set of five or more points in R3 must be affinely d...
 8.2.12E: Show that a set is affinely dependent when
 8.2.13E: Use only the definition of affine dependence to show that an indexe...
 8.2.14E: The conditions for affine dependence are stronger than those for li...
 8.2.15E: a. Show that the set S is affinely independent.b. Find the barycent...
 8.2.16E: a. Show that the set S is affinely independent.b. Find the barycent...
 8.2.17E: Prove Theorem 6 for an affinely independent set .[Hint: One method ...
 8.2.18E: Let T be a tetrahedron in “standard” position, with three edges alo...
 8.2.19E: be an affinely dependent set of points in Rn and let be a linear tr...
 8.2.20E: Suppose is an affinely independent set in Rn and q is an arbitrary ...
 8.2.21E: In Exercises 21–24, a, b, and c are noncollinear points in R2 and p...
 8.2.22E: In Exercises 21–24, a, b, and c are noncollinear points in R2 and p...
 8.2.23E: In Exercises 21–24, a, b, and c are noncollinear points in R2 and p...
 8.2.24E: In Exercises 21–24, a, b, and c are noncollinear points in R2 and p...
Solutions for Chapter 8.2: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 8.2
Get Full SolutionsChapter 8.2 includes 24 full stepbystep solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. 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: 5. Since 24 problems in chapter 8.2 have been answered, more than 42771 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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.