Algebra and Trigonometry: Real Mathematics, Real People 7th Edition - Solutions by Chapter

Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

The n roots are the eigenvalues of A.

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).

space of all combinations of the columns of A.

If diagonalizable, they share n eigenvectors.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = 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 .

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

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

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

P = aaT laTa has rank l.

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

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

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

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

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

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