 10.20.1: Determine whether the vector v is a convex combination of the vecto...
 10.20.2: Verify Equation 7 for the two triangulations given in Figure 10.20.4.
 10.20.3: Let an affine transformation be given by a 2 2 matrix M and a twod...
 10.20.4: (a) Exhibit a triangulation of the points in Figure 10.20.4 in whic...
 10.20.5: Find the matrix M and twodimensional vector b that define the affi...
 10.20.6: (a) Let a and b be linearly independent vectors in the plane. Show ...
 10.20.7: (a) What can you say about the coefficients , , and that determine ...
 10.20.8: (a) The centroid of a triangle lies on the line segment connecting ...
 10.20.T1: To warp or morph a surface in we must be able to triangulate the su...
 10.20.T2: To warp or morph a solid object in we first partition the object in...
Solutions for Chapter 10.20: Warps and Morphs
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 10.20: Warps and Morphs
Get Full SolutionsChapter 10.20: Warps and Morphs includes 10 full stepbystep solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. This expansive textbook survival guide covers the following chapters and their solutions. Since 10 problems in chapter 10.20: Warps and Morphs have been answered, more than 15434 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.

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