 11.1: For Exercises 14, solve each proportion.
 11.2: For Exercises 14, solve each proportion.
 11.3: For Exercises 14, solve each proportion.
 11.4: For Exercises 14, solve each proportion.
 11.5: In Exercises 5 and 6, measurements are in centimeters. ABCDE FGHIJ
 11.6: In Exercises 5 and 6, measurements are in centimeters. ABC DBA
 11.7: Application David is 5 ft 8 in. tall and wants to find the height o...
 11.8: A certain magnifying glass, when held 6 in. from an object, creates...
 11.9: Construction Construct KL. Then find a point P that divides KL into...
 11.10: If two triangles are congruent, are they similar? Explain.
 11.11: Patsy does a juggling act. She sits on a stool that sits on top of ...
 11.12: Charlie builds a rectangular box home for his pet python and uses 1...
 11.13: Suppose you had a real clothespin similar to the sculpture at right...
 11.14: The ratio of the perimeters of two similar parallelograms is 3:7. W...
 11.15: The ratio of the surface areas of two spheres is . What is the rati...
 11.16: Application The Jones family paid $150 to a painting contractor to ...
 11.17: The dimensions of the smaller cylinder are twothirds of the dimens...
 11.18: w x y
 11.19: Below is a 58foot statue of Bahubali, in Sravanabelagola, India. E...
 11.20: Greek mathematician Archimedes liked the design at right so much th...
 11.21: Many fanciful stories are about people who accidentally shrink to a...
 11.22: Would 15 pounds of 1inch ice cubes melt faster than a 15pound blo...
Solutions for Chapter 11: Similarity
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 11: Similarity
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CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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!).

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.

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

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.