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

Tv = Av + Vo = linear transformation plus shift.

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!

A = CTC = (L.J]))(L.J]))T for positive definite A.

B j has b replacing column j of A; x j = det B j I det A

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 identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

Columns without pivots; these are combinations of earlier columns.

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

A symmetric matrix with eigenvalues of both signs (+ and - ).

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.

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

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

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

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

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

One free variable is Si = 1, other free variables = o.

Spectral radius = max of IAi I.

Signs in A = signs in D.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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