 6.2.1: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.2: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.3: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.4: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.5: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.6: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.7: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.8: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.9: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.10: Solve Exercises 110. State which conjectures or definitions you used.
 6.2.11: Developing Proof Whats wrong with this picture?
 6.2.12: Developing Proof Whats wrong with this picture?
 6.2.13: Draw a circle and two chords of unequal length. Which is closer to ...
 6.2.14: Draw two circles with different radii. In each circle, draw a chord...
 6.2.15: Polygon MNOP is a rectangle inscribed in a circle centered at the o...
 6.2.16: Construction Construct a triangle. Using the sides of the triangle ...
 6.2.17: Construction Trace a circle onto a blank sheet of paper without usi...
 6.2.18: Construction Adventurer Dakota Davis digs up a piece of a circular ...
 6.2.19: Construction The satellite photo at right shows only a portion of a...
 6.2.20: Developing Proof Complete the flowchart proof shown, which proves t...
 6.2.21: Circle O has center (0, 0) and passes through points A(3, 4) and B(...
 6.2.22: Developing Proof Identify each of these statements as true or false...
 6.2.23: MiniInvestigation Use what you learned in the last lesson about th...
 6.2.24: Developing Proof Given that PA and PB are both tangent to circle Q ...
 6.2.25: Rachel and Yulia are building an art studio above their back bedroo...
 6.2.26: What will the units digit be when you evaluate 323 ?
 6.2.27: A small lightwing aircraft has made an emergency landing in a remo...
 6.2.28: Consider the figure at right with line AB . As P moves from left to...
Solutions for Chapter 6.2: Chord Properties
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 6.2: Chord Properties
Get Full SolutionsChapter 6.2: Chord Properties includes 28 full stepbystep solutions. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. This expansive textbook survival guide covers the following chapters and their solutions. Since 28 problems in chapter 6.2: Chord Properties have been answered, more than 22120 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

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

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

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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