- 8.6.1: In Exercises 18, find the area of the shaded region. The radius of ...
- 8.6.2: In Exercises 18, find the area of the shaded region. The radius of ...
- 8.6.3: In Exercises 18, find the area of the shaded region. The radius of ...
- 8.6.4: In Exercises 18, find the area of the shaded region. The radius of ...
- 8.6.5: In Exercises 18, find the area of the shaded region. The radius of ...
- 8.6.6: In Exercises 18, find the area of the shaded region. The radius of ...
- 8.6.7: In Exercises 18, find the area of the shaded region. The radius of ...
- 8.6.8: In Exercises 18, find the area of the shaded region. The radius of ...
- 8.6.9: The shaded area is 12 cm2. Find r
- 8.6.10: The shaded area is 32 cm2. Find r.
- 8.6.11: The shaded area is 120 cm2, and the radius is 24 cm. Find x.
- 8.6.12: The shaded area is 10 cm2 The radius of the large circle is 10 cm, ...
- 8.6.13: Application Suppose the pizza slice in the photo at the beginning o...
- 8.6.14: Utopia Park has just installed a circular fountain 8 meters in diam...
- 8.6.15: The illustrations below demonstrate how to find a rectangle with th...
- 8.6.16: Construction Reverse the process you used in Exercise 15. On graph ...
- 8.6.17: Each set of circles is externally tangent. What is the area of the ...
- 8.6.18: The height of a trapezoid is 15 m and the midsegment is 32 m. What ...
- 8.6.19: CE, BH,and AG are altitudes. Find AB and AG.
- 8.6.20: In Exercises 2023, identify each statement as true or false. If tru...
- 8.6.21: In Exercises 2023, identify each statement as true or false. If tru...
- 8.6.22: In Exercises 2023, identify each statement as true or false. If tru...
- 8.6.23: In Exercises 2023, identify each statement as true or false. If tru...
Solutions for Chapter 8.6: Any Way You Slice It
Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition
Tv = Av + Vo = linear transformation plus shift.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
peA) = det(A - AI) has peA) = zero matrix.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
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).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.