Precalculus With Limits A Graphing Approach 5th Edition - Solutions by Chapter

Upper triangular systems are solved in reverse order Xn to Xl.

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.

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

Columns without pivots; these are combinations of earlier columns.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Vector addition and scalar multiplication.

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.

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

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

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

The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Every B = M-I AM has the same eigenvalues as A.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

One free variable is Si = 1, other free variables = o.

Orthonormal columns (complex analog of Q).

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.