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Solutions for Chapter 5.7: Integration Using Tables and Computer Algebra Systems
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 5.7: Integration Using Tables and Computer Algebra Systems
Get Full SolutionsBiocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Chapter 5.7: Integration Using Tables and Computer Algebra Systems includes 28 full stepbystep solutions. Since 28 problems in chapter 5.7: Integration Using Tables and Computer Algebra Systems have been answered, more than 24951 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
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!

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Kirchhoff's Laws.
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.

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

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Norm
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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.