Algebra and Trigonometry 8th Edition - Solutions by Chapter

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

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.

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.

Columns without pivots; these are combinations of earlier columns.

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

Triangular matrix with one extra nonzero adjacent diagonal.

Blocks aij B, eigenvalues Ap(A)Aq(B).

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

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

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

Column and row spaces = lines cu and cv.

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

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 ().

Appears in block elimination on [~ g ].

The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

T- 1 has rank 1 above and below diagonal.