Explorations in College Algebra 5th Edition - Solutions by Chapter

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.

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).

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

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.

Use AT for complex A.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Complex analog a j i = aU of a symmetric matrix.

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.

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

Vector addition and scalar multiplication.

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

= Xl (column 1) + ... + xn(column n) = combination of columns.

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 •

= column times row = rank one matrix.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spectral radius = max of IAi I.

Constant down each diagonal = time-invariant (shift-invariant) filter.

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