Biocalculus: Calculus for Life Sciences 1st Edition - Solutions by Chapter

Parentheses can be removed to leave ABC.

Upper triangular systems are solved in reverse order Xn to Xl.

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!

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

Ax = AX with x#-O so det(A - AI) = o.

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

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

IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Appears in block elimination on [~ g ].

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

Spectral radius = max of IAi I.

Columns of n by n identity matrix (written i ,j ,k in R3).

Signs in A = signs in D.