 Chapter 1.1: The statement 3 + 3 = 6 serves as a/an ___________ to the conjectur...
 Chapter 1.2: Arriving at a specific conclusion from one or more general statemen...
 Chapter 1.3: Arriving at a general conclusion based on ob
 Chapter 1.4: True or False: A theorem cannot have counterexamples. ____
 Chapter 1.5: Fill in each blank so that the resulting statement is true.True or ...
 Chapter 1.6: Fill in each blank so that the resulting statement is true.True or ...
 Chapter 1.7: In 310, identify a pattern in each list of numbers. Then use this p...
 Chapter 1.8: In 310, identify a pattern in each list of numbers. Then use this p...
 Chapter 1.9: In 310, identify a pattern in each list of numbers. Then use this p...
 Chapter 1.10: In 310, identify a pattern in each list of numbers. Then use this p...
 Chapter 1.11: Identify a pattern in the following sequence of figures. Then use t...
 Chapter 1.12: In 1213, use inductive reasoning to predict the next line in each s...
 Chapter 1.13: In 1213, use inductive reasoning to predict the next line in each s...
 Chapter 1.14: Consider the following procedure: Select a number. Double the numbe...
 Chapter 1.15: The number 923,187,456 iscalledapandigitalsquarebecause it uses all...
 Chapter 1.16: A magnified view of the boundary of this black buglike shape, calle...
 Chapter 1.17: In 1720, obtain an estimate for each computation by rounding the nu...
 Chapter 1.18: In 1720, obtain an estimate for each computation by rounding the nu...
 Chapter 1.19: In 1720, obtain an estimate for each computation by rounding the nu...
 Chapter 1.20: In 1720, obtain an estimate for each computation by rounding the nu...
 Chapter 1.21: Estimate the total cost of six grocery items if their prices are $8...
 Chapter 1.22: Estimate the salary of a worker who works for 78 hours at $6.85 per...
 Chapter 1.23: At a yard sale, a person bought 21 books at $0.85 each, two chairs ...
 Chapter 1.24: The circle graph shows how the 17,487,475 students enrolled in U.S....
 Chapter 1.25: A small private school employs 10 teachers with salaries ranging fr...
 Chapter 1.26: Select the best estimate for the number of seconds in a day. a. 150...
 Chapter 1.27: Imagine the entire global population as a village of precisely 200 ...
 Chapter 1.28: The bar graph shows the percentage of people 25 years of age and ol...
 Chapter 1.29: During a diagnostic evaluation, a 33yearold woman experienced a p...
 Chapter 1.30: The bar graph shows the population of the United States, in million...
 Chapter 1.31: What necessary piece of information is missing that prevents solvin...
 Chapter 1.32: In the following problem, there is one more piece of information gi...
 Chapter 1.33: If there are seven frankfurters in one pound, how many pounds would...
 Chapter 1.34: A car rents for $175 per week plus $0.30 per mile. Find the rental ...
 Chapter 1.35: You are choosing between two plans at a discount warehouse. Plan A ...
 Chapter 1.36: Miami is on Eastern Standard Time and San Francisco is on Pacific S...
 Chapter 1.37: An automobile purchased for $37,000 is worth $2600 after eight year...
 Chapter 1.38: Suppose you are an engineer programming the automatic gate for a 35...
Solutions for Chapter Chapter 1: Problem Solving and Critical Thinking
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter Chapter 1: Problem Solving and Critical Thinking
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Since 38 problems in chapter Chapter 1: Problem Solving and Critical Thinking have been answered, more than 61314 students have viewed full stepbystep solutions from this chapter. Chapter Chapter 1: Problem Solving and Critical Thinking includes 38 full stepbystep solutions. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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