- Chapter 1: Speaking Mathematically
- Chapter 10: Graphs and Trees
- Chapter 2: THE LOGIC OF COMPOUND STATEMENTS
- Chapter 3: The Logic of Quantied Statements
- Chapter 4: Elementary Number Theory and Methods of Proof
- Chapter 5: Sequences, Mathematical Induction, and Recursion
- Chapter 6: Set Theory
- Chapter 7: Functions
- Chapter 8: Relations
- Chapter 9: Counting and Probability
Discrete Mathematics: Introduction to Mathematical Reasoning 1st Edition - Solutions by Chapter
Full solutions for Discrete Mathematics: Introduction to Mathematical Reasoning | 1st Edition
Discrete Mathematics: Introduction to Mathematical Reasoning | 1st Edition - Solutions by ChapterGet Full Solutions
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.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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).
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
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.
A directed graph that has constants Cl, ... , Cm associated with the edges.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Solvable system Ax = b.
The right side b is in the column space of A.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Constant down each diagonal = time-invariant (shift-invariant) filter.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).
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