 8.3.1E: In R2, let S = Describe (or sketch) the convex hull of S.
 8.3.2E: Describe the convex hull of the set S of points2that satisfy the gi...
 8.3.3E: Consider the points in Exercise 5 in Section 8.1. Which of p1, p2, ...
 8.3.4E: Consider the points in Exercise 6 in Section 8.1. Which of p1, p2, ...
 8.3.5E: Determine whether p1 and p2 are in conv S.
 8.3.6E:
 8.3.7E: Exercises 7–10 use the terminology from Section 8.2.a. Let T = P1 =...
 8.3.8E: Exercises 7–10 use the terminology from Section 8.2. Reference Exer...
 8.3.9E: Exercises 7–10 use the terminology from Section 8.2.
 8.3.10E: Exercises 7–10 use the terminology from Section 8.2.Repeat Exercise...
 8.3.11E: In Exercises 11 and 12, mark each statement True or False. Justify ...
 8.3.12E: In Exercises 11 and 12, mark each statement True or False. Justify ...
 8.3.13E: Let S be a convex subset of Rn and suppose that is a linear transfo...
 8.3.14E: be a linear transformation and let T be a convex subset of Rm. Prov...
 8.3.15E: Use the procedure in the proof of Caratheodory’s Theorem to express...
 8.3.16E: Repeat Exercise 9 for points Reference Exercise 9:Exercises 7–10 us...
 8.3.17E: In Exercises 17–20, prove the given statement about subsets A and B...
 8.3.18E: In Exercises 17–20, prove the given statement about subsets A and B...
 8.3.19E: In Exercises 17–20, prove the given statement about subsets A and B...
 8.3.20E: In Exercises 17–20, prove the given statement about subsets A and B...
 8.3.21E:
 8.3.22E: Repeat Exercise 21 for Reference Exercise 21:
 8.3.23E: Let g(t) be defined as in Exercise 21. Its graph is called a quadra...
 8.3.24E: Given control points lies in the convex hull of the four control po...
Solutions for Chapter 8.3: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 8.3
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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.

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

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.