- 7.1: Express the ratio in simplest form
- 7.2: Express the ratio in simplest form
- 7.3: Express the ratio in simplest form
- 7.4: Express the ratio in simplest form
- 7.5: Is the ratio a:b always, sometimes, or never equal to the ratio b:a...
- 7.6: An office copy machine can make a reduction to 90%. thus making the...
- 7.7: Barbara is making oatmeal for breakfast. The instructions say to us...
- 7.8: Express the ratio in simplest form. Dl.IS
- 7.9: Express the ratio in simplest form. ST.DI
- 7.10: Express the ratio in simplest form. IT.DT
- 7.11: Express the ratio in simplest form. DI-.1T
- 7.12: Express the ratio in simplest form. 1T.DS
- 7.13: Express the ratio in simplest form. IS:DI:IT
- 7.14: What is the ratio of 750 mL to 1.5 L?
- 7.15: Can you find the ratio of 2 L to 4 km? Explain.
- 7.16: The ratio of the lengths of two segments is 4:3 when they are measu...
- 7.17: Three numbers aren't known, but the ratio of the numbers is 1 : 2: ...
- 7.18: What is the second term of the proportion - = -?
- 7.19: Exercises 15-20 refer to a triangle. Express the ratio of the heigh...
- 7.20: Exercises 15-20 refer to a triangle. Express the ratio of the heigh...
- 7.21: Write the algebraic ratio in simplest form. laAab
- 7.22: Write the algebraic ratio in simplest form. led5c2
- 7.23: Write the algebraic ratio in simplest form. 3U + 4)a(x + 4)
- 7.24: The ratio of the measures of two complementary angles is 4:5.
- 7.25: The ratio of the measures of two supplementary angles is 11:4.
- 7.26: The measures of the angles of a triangle are in the ratio 3:4:5.
- 7.27: The measures of the acute angles of a right triangle are in the rat...
- 7.28: The measures of the angles of an isosceles triangle are in the rati...
- 7.29: The measures of the angles of a hexagon are in the ratio 4:5:5:8:9:9.
- 7.30: The perimeter of a triangle is 132 cm and the lengths of its sides ...
- 7.31: The measures of the consecutive angles of a quadrilateral are in th...
- 7.32: What is the ratio of the measure of an interior angle to the measur...
- 7.33: A team's best hitter has a lifetime batting average of .320. He has...
- 7.34: A basketball player has made 24 points out of30 free throws. She ho...
- 7.35: Points B and C lie on AD. Find AC ifAB _ 3 ACBD ~ 4' CD-, and BD = 66.
- 7.36: Find the ratio of x to v: I- -v .v11 _ 1V X= 44= 44
- 7.37: Find the value of x.
- 7.38: Find the value of x.
- 7.39: Find the values of x and y.
- 7.40: Find the values of x and y.
- 7.41: Prove: If - = - = -, thenb d f b + d + fx - v +(////?/: Let - = r. ...
- 7.42: Explain how to extend the proof of Exercise 41 to justify Property ...
- 7.43: IfAa - 9b a - 2bAafind the numerical value of the ratio a: b.
Solutions for Chapter 7: Ratio and Proportion
Full solutions for Geometry | 1st Edition
Tv = Av + Vo = linear transformation plus shift.
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).
Column space C (A) =
space of all combinations of the columns of A.
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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).
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.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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.