Elementary Linear Algebra with Applications 9th Edition - Solutions by Chapter

Tv = Av + Vo = linear transformation plus shift.

Parentheses can be removed to leave ABC.

The n roots are the eigenvalues of A.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

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].

Columns without pivots; these are combinations of earlier columns.

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

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.

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

The columns of N are the n - r special solutions to As = 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).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

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

A square matrix that has no inverse: det(A) = o.

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

Signs in A = signs in D.

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.