 14.1.1: The graph models the baseball schedule for a week. The vertices rep...
 14.1.2: The graph models the baseball schedule for a week. The vertices rep...
 14.1.3: The graph models the baseball schedule for a week. The vertices rep...
 14.1.4: The graph models the baseball schedule for a week. The vertices rep...
 14.1.5: In 56, draw two equivalent graphs for each description.The vertices...
 14.1.6: In 56, draw two equivalent graphs for each description. The vertice...
 14.1.7: In 78, explain why the two figures show equivalent graphs. Then dra...
 14.1.8: In 78, explain why the two figures show equivalent graphs. Then dra...
 14.1.9: Eight students form a math homework group. The students in the grou...
 14.1.10: An environmental action group has six members, A, B, C, D, E, and F...
 14.1.11: In 1112, draw a graph that models the layout of the city shown in e...
 14.1.12: In 1112, draw a graph that models the layout of the city shown in e...
 14.1.13: In 1314, create a graph that models the bordering relationships amo...
 14.1.14: In 1314, create a graph that models the bordering relationships amo...
 14.1.15: In 1518, draw a graph that models the connecting relationships in e...
 14.1.16: In 1518, draw a graph that models the connecting relationships in e...
 14.1.17: In 1518, draw a graph that models the connecting relationships in e...
 14.1.18: In 1518, draw a graph that models the connecting relationships in e...
 14.1.19: In 1920, a security guard needs to walk the streets of the neighbor...
 14.1.20: In 1920, a security guard needs to walk the streets of the neighbor...
 14.1.21: In 2122, a mail carrier is to walk the streets of the neighborhood ...
 14.1.22: n 2333, use the following graph. B A C D F
 14.1.23: In 2333, use the following graph. B A C D F
 14.1.24: In 2333, use the following graph. B A C D F
 14.1.25: In 2333, use the following graph. B A C D F
 14.1.26: In 2333, use the following graph. B A C D F
 14.1.27: In 2333, use the following graph. B A C D F
 14.1.28: In 2333, use the following graph. B A C D F
 14.1.29: In 2333, use the following graph. B A C D F
 14.1.30: In 2333, use the following graph. B A C D F
 14.1.31: In 2333, use the following graph. B A C D F
 14.1.32: In 2333, use the following graph. B A C D F
 14.1.33: In 2333, use the following graph. B A C D F
 14.1.34: In 3448, use the following graph.
 14.1.35: In 3448, use the following graph.
 14.1.36: In 3448, use the following graph.
 14.1.37: In 3448, use the following graph.
 14.1.38: In 3448, use the following graph.
 14.1.39: In 3448, use the following graph.
 14.1.40: In 3448, use the following graph.
 14.1.41: In 3448, use the following graph.
 14.1.42: In 3448, use the following graph.
 14.1.43: In 3448, use the following graph.
 14.1.44: In 3448, use the following graph.
 14.1.45: In 3448, use the following graph.
 14.1.46: In 3448, use the following graph.
 14.1.47: In 3448, use the following graph.
 14.1.48: In 3448, use the following graph.
 14.1.49: In 4952, draw a graph with the given characteristics.The graph has ...
 14.1.50: In 4952, draw a graph with the given characteristics.The graph has ...
 14.1.51: In 4952, draw a graph with the given characteristics.The graph has ...
 14.1.52: In 4952, draw a graph with the given characteristics.The graph has ...
 14.1.53: What is a graph? Define vertices and edges as part of your descript...
 14.1.54: Describe how to determine whether or not a point where two of a gra...
 14.1.55: What are equivalent graphs?
 14.1.56: Because a graph can be drawn in many equivalent ways, describe the ...
 14.1.57: What is meant by the degree of a graphs vertex and how is it determ...
 14.1.58: Describe how to determine if a vertex is even or odd.
 14.1.59: What are adjacent vertices? If two vertices are near each other in ...
 14.1.60: What is a path in a graph?
 14.1.61: What is a circuit? Describe the difference between a path and a cir...
 14.1.62: What is a connected graph?
 14.1.63: What is a bridge?
 14.1.64: Describe a situation involving relationships that can be modeled wi...
 14.1.65: Describe one advantage and one disadvantage of using the graph for ...
 14.1.66: Make Sense? In 6669, determine whether each statement makes sense o...
 14.1.67: Make Sense? In 6669, determine whether each statement makes sense o...
 14.1.68: Make Sense? In 6669, determine whether each statement makes sense o...
 14.1.69: Make Sense? In 6669, determine whether each statement makes sense o...
 14.1.70: Draw a graph with six vertices and two bridges
 14.1.71: Use the information in Exercise 10 to draw a graph that models whic...
 14.1.72: Use inductive reasoning to make a conjecture that compares the sum ...
 14.1.73: Group members should determine a relationship that exists among som...
 14.1.74: Each group member should select a favorite television series, movie...
Solutions for Chapter 14.1: Graphs, Paths, and Circuits
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter 14.1: Graphs, Paths, and Circuits
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Chapter 14.1: Graphs, Paths, and Circuits includes 74 full stepbystep solutions. Since 74 problems in chapter 14.1: Graphs, Paths, and Circuits have been answered, more than 62737 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Lucas numbers
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.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.