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# Solutions for Chapter 2: Logic

## Full solutions for Mathematical Proofs: A Transition to Advanced Mathematics | 3rd Edition

ISBN: 9780321797094

Solutions for Chapter 2: Logic

Solutions for Chapter 2
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##### ISBN: 9780321797094

Since 100 problems in chapter 2: Logic have been answered, more than 5712 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Mathematical Proofs: A Transition to Advanced Mathematics, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Mathematical Proofs: A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780321797094. Chapter 2: Logic includes 100 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Big formula for n by n determinants.

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• 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).

• Column space C (A) =

space of all combinations of the columns of A.

• 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.

• 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.

• Graph G.

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.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• 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.

• Norm

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.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• 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.

• Simplex method for linear programming.

The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

• Singular matrix A.

A square matrix that has no inverse: det(A) = o.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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