- 1-3.1: Perform each indicated operation.1058 - 3110
- 1-3.2: Perform each indicated operation.34,18
- 1-3.3: Perform each indicated operation.5 - 1-42 + 1-22
- 1-3.4: Perform each indicated operation.1-322 - 1-4212425122 - 1-223
- 1-3.5: Perform each indicated operation.True or false?413 - 922 - 6 6
- 1-3.6: Perform each indicated operation.Find the value of a xzwhen x = -2,...
- 1-3.7: Perform each indicated operation.What property does 31-2 + x2 = -6 ...
- 1-3.8: Perform each indicated operation.Simplify -4p - 6 + 3p + 8 by combi...
- 1-3.9: Solve.V = 13 pr2 h for h
- 1-3.10: Solve.6 - 311 + x2 = 21x + 52 - 2
- 1-3.11: Solve.-1m - 32 = 5 - 2m
- 1-3.12: Solve.x - 23 = 2x + 15
- 1-3.13: Solve each inequality, and graph the solution set.-2.5x 6 6.5
- 1-3.14: Solve each inequality, and graph the solution set.41x + 32 - 5x 6 12
- 1-3.15: Solve each inequality, and graph the solution set.23x - 16 -2
- 1-3.16: Solve each problem.The gap in average annual earnings by level ofed...
- 1-3.17: Solve each problem.Baby boomers are expected to inherit$10.4 trilli...
- 1-3.18: Solve each problem.Use the answer from Exercise 17(b) to estimateth...
- 1-3.19: Consider the linear equation Find the following.The x- and y-interc...
- 1-3.20: Consider the linear equation Find the following.The slope
- 1-3.21: Consider the linear equation Find the following.The graph
- 1-3.22: Are the lines with equations and parallel, perpendicular, or neither?
- 1-3.23: Write an equation for each line. Give the final answer in slope-int...
- 1-3.24: Write an equation for each line. Give the final answer in slope-int...
Solutions for Chapter 1-3: Linear Equations and Inequalities in Two Variables; Functions
Full solutions for Beginning Algebra | 11th Edition
Solutions for Chapter 1-3: Linear Equations and Inequalities in Two Variables; FunctionsGet Full Solutions
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).
Upper triangular systems are solved in reverse order Xn to Xl.
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).
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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].
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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.
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.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.