Elementary and Intermediate Algebra 5th Edition - Solutions by Chapter

Tv = Av + Vo = linear transformation plus shift.

space of all combinations of the columns of A.

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.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

Columns without pivots; these are combinations of earlier columns.

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 .

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

= column times row = rank one matrix.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

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

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

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

Any vector space inside V, including V and Z = {zero vector only}.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

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