 1.2.1E: Which of the following sets are equal?A = {a, b, c, d}_____________...
 1.2.2E: Write in words how to read each of the following out loud.a. {x ? R...
 1.2.3E: a. Is 4 = {4}?b. How many elements are in the set {3, 4, 3, 5}?c. H...
 1.2.4E: a. Is 2 ? {2}?b. How many elements are in the set {2, 2, 2, 2}?c. H...
 1.2.5E: Which of the following sets are equal?A = {0, 1, 2}B = {x ? R  –1 ...
 1.2.6E: For each integer n, let Tn = {n, n2}. How many elements are in each...
 1.2.7E: Use the setroster notation to indicate the elements in each of the...
 1.2.8E: Let A = {c, d, f, g}, B = {f, j}, and C = {d, g}. Answer each of th...
 1.2.9E: a. Is 3 ? {1, 2, 3}?b. Is 1 ? {1}?c. Is {2} ? {1, 2}?d. Is {3} ? {1...
 1.2.10E: a. Is ((–2)2, –22) = (–22, (–2)2)?b. Is (5, –5) = (–5, 5)?c. Is (8 ...
 1.2.11E: Let A = {w, x, y, z} and B = {a, b}. Use the setroster notation to...
 1.2.12E: Let S = {2, 4, 6} and T = {1,3, 5}. Use the setroster notation to ...
Solutions for Chapter 1.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 1.2
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 12 problems in chapter 1.2 have been answered, more than 45242 students have viewed full stepbystep solutions from this chapter. Chapter 1.2 includes 12 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Outer product uv T
= column times row = rank one matrix.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.