Solutions for Chapter 9: Systems of Linear Equations and Inequalities

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!

The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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

Invert A by row operations on [A I] to reach [I A-I].

Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Triangular matrix with one extra nonzero adjacent diagonal.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

A sequence of steps intended to approach the desired solution.

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

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

Vector addition and scalar multiplication.

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).

Every v in V is orthogonal to every w in W.

P = aaT laTa has rank l.

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

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

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.