Algebra and Trigonometry 8th Edition - Solutions by Chapter

Remove row i and column j; multiply the determinant by (-I)i + j •

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

dim(V) = number of vectors in any basis for V.

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.

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

Constant along each antidiagonal; hij depends on i + j.

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

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

Any solution to Ax = b; often x p has free variables = o.

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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).

Column vectors by convention.

The right side b is in the column space of A.

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

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

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

T- 1 has rank 1 above and below diagonal.

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.

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