 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: Let v1 = v2 = v3 = {v1, v2, v3}.a. Show that the set S is affinely ...
 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 {p1, p2, p3} is an affinely independent set in Rn and q is ...
 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 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 8.2
Get Full SolutionsChapter 8.2 includes 24 full stepbystep solutions. Since 24 problems in chapter 8.2 have been answered, more than 32653 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: 9780321385178. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4.

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

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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

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