Advanced Engineering Mathematics 6th Edition - Solutions by Chapter

Parentheses can be removed to leave ABC.

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).

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.

B j has b replacing column j of A; x j = det B j I det A

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

dim(V) = number of vectors in any basis for V.

Columns without pivots; these are combinations of earlier columns.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

A sequence of steps intended to approach the desired solution.

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.

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

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

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.

T- 1 has rank 1 above and below diagonal.

Orthonormal columns (complex analog of Q).

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).