Precalculus With Limits A Graphing Approach 5th 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!

peA) = det(A - AI) has peA) = zero matrix.

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).

If diagonalizable, they share n eigenvectors.

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

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.

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

A sequence of steps intended to approach the desired solution.

Square root of x T x (Pythagoras in n dimensions).

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).

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).

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

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

Column vectors by convention.

A square matrix that has no inverse: det(A) = o.

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

For matrix norms II A + B II < II A II + II B II·

The rows (or the columns) of A generate a box with volume I det(A) I.