Advanced Engineering Mathematics 6th Edition - Solutions by Chapter

Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

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

cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Columns without pivots; these are combinations of earlier columns.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

A directed graph that has constants Cl, ... , Cm associated with the edges.

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

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

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

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

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.