Algebra and Trigonometry with Analytic Geometry 12th 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).

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!

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

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

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

Ax = AX with x#-O so det(A - AI) = o.

Columns without pivots; these are combinations of earlier columns.

Constant along each antidiagonal; hij depends on i + j.

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

A symmetric matrix with eigenvalues of both signs (+ and - ).

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

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

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

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

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

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Orthonormal columns (complex analog of Q).

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.