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

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

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

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].

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

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

A sequence of steps intended to approach the desired solution.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

Nullspace of AT = "left nullspace" of A because y T A = OT.

Vector addition and scalar multiplication.

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).

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

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

Orthogonal Q times positive (semi)definite H.

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.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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