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# Solutions for Chapter 9.2: Partial Derivatives

## Full solutions for Biocalculus: Calculus for Life Sciences | 1st Edition

ISBN: 9781133109631

Solutions for Chapter 9.2: Partial Derivatives

Solutions for Chapter 9.2
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##### ISBN: 9781133109631

This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Since 73 problems in chapter 9.2: Partial Derivatives have been answered, more than 27182 students have viewed full step-by-step solutions from this chapter. Chapter 9.2: Partial Derivatives includes 73 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Augmented matrix [A b].

Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Column space C (A) =

space of all combinations of the columns of A.

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Dimension of vector space

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

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Identity matrix I (or In).

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

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Markov matrix M.

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.

• Multiplication Ax

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

• Normal equation AT Ax = ATb.

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

• Outer product uv T

= column times row = rank one matrix.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Similar matrices A and B.

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

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Toeplitz matrix.

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

• Transpose matrix AT.

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