Contemporary Abstract Algebra 8th Edition - Solutions by Chapter

peA) = det(A - AI) has peA) = zero matrix.

space of all combinations of the columns of A.

If diagonalizable, they share n eigenvectors.

A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

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

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Constant along each antidiagonal; hij depends on i + j.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

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

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

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

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

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.