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# Solutions for Chapter 3.2: Discrete Mathematics and Its Applications 7th Edition

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

ISBN: 9780073383095

Solutions for Chapter 3.2

Solutions for Chapter 3.2
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##### ISBN: 9780073383095

This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.2 includes 74 full step-by-step solutions. Since 74 problems in chapter 3.2 have been answered, more than 124789 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Column space C (A) =

space of all combinations of the columns of A.

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

• 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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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).

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• Multiplication Ax

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

• Normal equation AT Ax = ATb.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

• Orthogonal subspaces.

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

• 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.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• 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.

• Schwarz inequality

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

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

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

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

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