- Chapter 1: Linear Equations
- Chapter 1.1: Lines
- Chapter 1.2: Pairs of Lines
- Chapter 1.3: Applications in Business and Economics
- Chapter 1.4: Scatter Diagrams; Linear Curve Fitting
- Chapter 10: Markov Chains; Games
- Chapter 10.1: Markov Chains and Transition Matrices
- Chapter 10.2: Regular Markov Chains
- Chapter 10.3: Absorbing Markov Chains
- Chapter 10.4: Two-Person Games
- Chapter 10.5: Mixed Strategies
- Chapter 10.6: Optimal Strategy in Two-Person Zero-Sum Games with 2 2 Matrices
- Chapter 11: Logic
- Chapter 11.1: Propositions
- Chapter 11.2: Truth Tables
- Chapter 11.3: Implications;The Biconditional Connective;Tautologies
- Chapter 11.4: Arguments
- Chapter 11.5: Logic Circuits
- Chapter 2: Systems of Linear Equations
- Chapter 2.1: Systems of Linear Equations: Substitution; Elimination
- Chapter 2.2: Systems of Linear Equations: Gaussian Elimination
- Chapter 2.3: Systems of m Linear Equations Containing n Variables
- Chapter 3: Matrices
- Chapter 3.1: Matrix Algebra
- Chapter 3.2: Multiplication of Matrices
- Chapter 3.3: The Inverse of a Matrix
- Chapter 3.4: Applications in Economics (the Leontief Model), Accounting, and Statistics (the Method of Least Squares)*
- Chapter 4: Linear Programming with Two Variables
- Chapter 4.1: Systems of Linear Inequalities
- Chapter 4.2: A Geometric Approach to Linear Programming Problems with Two Variables*
- Chapter 4.3: Models Utilizing Linear Programming with Two Variables
- Chapter 5: Linear Programming: Simplex Method
- Chapter 5.1: The Simplex Tableau; Pivoting
- Chapter 5.2: The Simplex Method: Solving Maximum Problems in Standard Form
- Chapter 5.3: Solving Minimum Problems Using the Duality Principle
- Chapter 5.4: The Simplex Method for Problems Not in Standard Form
- Chapter 6: Finance
- Chapter 6.1: Interest
- Chapter 6.2: Compound Interest
- Chapter 6.3: Annuities; Sinking Funds
- Chapter 6.4: Present Value of an Annuity; Amortization
- Chapter 6.5: Annuities and Amortization Using Recursive Sequences
- Chapter 7: Probability
- Chapter 7.1: Sets
- Chapter 7.2: The Number of Elements in a Set
- Chapter 7.3: The Multiplication Principle
- Chapter 7.4: Sample Spaces and the Assignment of Probabilities
- Chapter 7.5: Properties of the Probability of an Event
- Chapter 7.6: Expected Value
- Chapter 8: Additional Probability Topics
- Chapter 8.1: Conditional Probability
- Chapter 8.2: Independent Events
- Chapter 8.3: Bayes Theorem
- Chapter 8.4: Permutations
- Chapter 8.5: Combinations
- Chapter 8.6: The Binomial Probability Model
- Chapter 9: Statistics
- Chapter 9.1: Introduction to Statistics: Data and Sampling
- Chapter 9.2: Representing Qualitative Data Graphically: Bar Graphs; Pie Charts
- Chapter 9.3: Organizing and Displaying Quantitative Data
- Chapter 9.4: Measures of Central Tendency
- Chapter 9.5: Measures of Dispersion
- Chapter 9.6: The Normal Distribution
Finite Mathematics, Binder Ready Version: An Applied Approach 11th Edition - Solutions by Chapter
Full solutions for Finite Mathematics, Binder Ready Version: An Applied Approach | 11th Edition
Finite Mathematics, Binder Ready Version: An Applied Approach | 11th Edition - Solutions by ChapterGet Full Solutions
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.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
A = CTC = (L.J]))(L.J]))T for positive definite 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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
A sequence of steps intended to approach the desired solution.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Outer product uv T
= column times row = rank one matrix.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rank r (A)
= number of pivots = dimension of column space = dimension of row 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).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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