California Algebra 2: Concepts, Skills, and Problem Solving 1st Edition - Solutions by Chapter

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.)

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

A sequence of steps intended to approach the desired solution.

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Blocks aij B, eigenvalues Ap(A)Aq(B).

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

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

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.

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).

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Every B = M-I AM has the same eigenvalues as A.

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

Spectral radius = max of IAi I.

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

T- 1 has rank 1 above and below diagonal.

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