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# Solutions for Chapter 6.2: Discrete Probability

## Full solutions for Discrete Mathematics and Its Applications | 6th Edition

ISBN: 9780073229720

Solutions for Chapter 6.2: Discrete Probability

Solutions for Chapter 6.2
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##### ISBN: 9780073229720

This expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 6.2: Discrete Probability have been answered, more than 36035 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Chapter 6.2: Discrete Probability includes 41 full step-by-step solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720.

Key Math Terms and definitions covered in this textbook
• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• 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.

• Column space C (A) =

space of all combinations of the columns of A.

• Diagonal matrix D.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Indefinite matrix.

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

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• 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.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Nullspace N (A)

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

• Orthogonal subspaces.

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

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

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