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# Solutions for Chapter 10.1: Discrete Mathematics and Its Applications 7th Edition

## Full solutions for Discrete Mathematics and Its Applications | 7th Edition

ISBN: 9780073383095

Solutions for Chapter 10.1

Solutions for Chapter 10.1
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##### ISBN: 9780073383095

Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 36 problems in chapter 10.1 have been answered, more than 221526 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.1 includes 36 full step-by-step solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Key Math Terms and definitions covered in this textbook
• Block matrix.

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.

• Change of basis matrix M.

The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

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

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

• Complex conjugate

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

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Hilbert matrix hilb(n).

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• Lucas numbers

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.

• Matrix multiplication AB.

The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

• 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.

• Partial pivoting.

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

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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