Linear Algebra with Applications 4th 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 vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

B j has b replacing column j of A; x j = det B j I det A

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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].

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

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

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

IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Orthogonal Q times positive (semi)definite H.

Each equation gives a plane in Rn; the planes intersect at x.

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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!

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

Spectral radius = max of IAi I.