Explorations in College Algebra 5th 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).

space of all combinations of the columns of A.

If diagonalizable, they share n eigenvectors.

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.

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

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

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

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

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).

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.

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

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

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

Appears in block elimination on [~ g ].

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

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

Orthonormal columns (complex analog of Q).

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