Discrete Mathematics with Applications 4th Edition - Solutions by Chapter

Parentheses can be removed to leave ABC.

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.)

cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

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

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

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.

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

has derivative AeAt; eAt u(O) solves u' = Au.

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.

Vector addition and scalar multiplication.

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

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

= number of pivots = dimension of column space = dimension of row space.

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

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

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

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

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