College Algebra 7th Edition - Solutions by Chapter

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 must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

Columns without pivots; these are combinations of earlier columns.

Invert A by row operations on [A I] to reach [I A-I].

Triangular matrix with one extra nonzero adjacent diagonal.

Diagonal entries = 1, off-diagonal entries = 0.

A symmetric matrix with eigenvalues of both signs (+ and - ).

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.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

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

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

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

Column vectors by convention.

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!

(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Real symmetric A has real A'S and orthonormal q's.

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

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.