Solutions for Chapter 17: Contemporary Abstract Algebra 8th Edition

Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

space of all combinations of the columns of A.

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

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.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Complex analog a j i = aU of a symmetric matrix.

Triangular matrix with one extra nonzero adjacent diagonal.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Nullspace of AT = "left nullspace" of A because y T A = OT.

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

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

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.

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

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 ().

Spectral radius = max of IAi I.