Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) 3rd Edition - Solutions by Chapter

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!

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

Columns without pivots; these are combinations of earlier columns.

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

Diagonal entries = 1, off-diagonal entries = 0.

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

A combination other than all Ci = 0 gives L Ci Vi = O.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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).

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

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.

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

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}.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Orthonormal columns (complex analog of Q).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).