Algebra and Trigonometry with Analytic Geometry 12th Edition - Solutions by Chapter

Upper triangular systems are solved in reverse order Xn to Xl.

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 matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

dim(V) = number of vectors in any basis for V.

Independent columns, N(A) = {O}, no free variables.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

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

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

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

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

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

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Any vector space inside V, including V and Z = {zero vector only}.

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