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# Solutions for Chapter 11: Sets I: Introduction, Subsets

## Full solutions for Mathematics: A Discrete Introduction | 3rd Edition

ISBN: 9780840049421

Solutions for Chapter 11: Sets I: Introduction, Subsets

Solutions for Chapter 11
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##### ISBN: 9780840049421

This textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3. Since 8 problems in chapter 11: Sets I: Introduction, Subsets have been answered, more than 8842 students have viewed full step-by-step solutions from this chapter. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421. Chapter 11: Sets I: Introduction, Subsets includes 8 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their 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.

• 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).

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• 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.

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Gram-Schmidt orthogonalization A = QR.

Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Random matrix rand(n) or randn(n).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• 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!

• 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.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B IIĀ·

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.

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