Trigonometry 11th Edition - Solutions by Chapter

A = CTC = (L.J]))(L.J]))T for positive definite A.

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

has derivative AeAt; eAt u(O) solves u' = Au.

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Orthogonal Q times positive (semi)definite H.

P = aaT laTa has rank l.

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

Column vectors by convention.

Appears in block elimination on [~ g ].

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

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!

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

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