- Chapter 1: Joy
- Chapter 10: Sets I: Introduction, Subsets
- Chapter 11: Sets I: Introduction, Subsets
- Chapter 12: Sets II: Operations
- Chapter 13: Combinatorial Proof: Two Examples
- Chapter 14: Relations
- Chapter 15: Equivalence Relations
- Chapter 16: Partitions
- Chapter 17: Binomial Coefficients
- Chapter 18: Counting Multisets
- Chapter 19: Inclusion-Exclusion
- Chapter 2: Speaking (and Writing) of Mathematics
- Chapter 20: Contradiction
- Chapter 21: Smallest Counterexample
- Chapter 22: Induction
- Chapter 23: Recurrence Relations
- Chapter 24: Functions
- Chapter 25: The Pigeonhole Principle
- Chapter 26: Composition
- Chapter 27: Permutations
- Chapter 28: Symmetry
- Chapter 29: Assorted Notation
- Chapter 3: Definition
- Chapter 30: Sample Space
- Chapter 31: Events
- Chapter 32: Conditional Probability and Independence
- Chapter 33: Random Variables
- Chapter 34: Expectation
- Chapter 35: Dividing
- Chapter 36: Greatest Common Divisor
- Chapter 37: Modular Arithmetic
- Chapter 38: The Chinese Remainder Theorem
- Chapter 39: Factoring
- Chapter 4: Theorem
- Chapter 40: Groups
- Chapter 41: Group Isomorphism The Same?
- Chapter 42: Subgroups
- Chapter 43: Fermats Little Theorem
- Chapter 44: Public Key Cryptography I: Introduction The Problem: Private Communication in Public
- Chapter 45: Public Key Cryptography II: Rabins Method
- Chapter 46: Public Key Cryptography III: RSA
- Chapter 47: Fundamentals of Graph Theory
- Chapter 48: Subgraphs
- Chapter 49: Connection
- Chapter 5: Proof
- Chapter 50: Trees
- Chapter 51: Eulerian Graphs
- Chapter 52: Coloring
- Chapter 53: Planar Graphs
- Chapter 54: Fundamentals of Partially Ordered Sets
- Chapter 55: Max and Min
- Chapter 56: Linear Orders
- Chapter 57: Linear Extensions
- Chapter 58: Dimension
- Chapter 59: Lattices
- Chapter 6: Counterexample
- Chapter 7: Boolean Algebra
- Chapter 8: Lists
- Chapter 9: Factorial
- Chapter Chapter 1: Fundamentals
- Chapter Chapter 10: Partially Ordered Sets
- Chapter Chapter 2: Collections
- Chapter Chapter 3: Counting and Relations
- Chapter Chapter 4: More Proof
- Chapter Chapter 5: Functions
- Chapter Chapter 6: Probability
- Chapter Chapter 7: Number Theory
- Chapter Chapter 8: Algebra
- Chapter Chapter 9: Graphs
Mathematics: A Discrete Introduction 3rd Edition - Solutions by Chapter
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Tv = Av + Vo = linear transformation plus shift.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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.
Every v in V is orthogonal to every w in W.
Outer product uv T
= column times row = rank one matrix.
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).
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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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