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# Solutions for Chapter 10.2: Qualitative Analysis of Linear Systems

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

ISBN: 9781133109631

Solutions for Chapter 10.2: Qualitative Analysis of Linear Systems

Solutions for Chapter 10.2
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##### ISBN: 9781133109631

This expansive textbook survival guide covers the following chapters and their solutions. Since 39 problems in chapter 10.2: Qualitative Analysis of Linear Systems have been answered, more than 27336 students have viewed full step-by-step solutions from this chapter. Chapter 10.2: Qualitative Analysis of Linear Systems includes 39 full step-by-step solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1.

Key Math Terms and definitions covered in this textbook
• Back substitution.

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

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Column space C (A) =

space of all combinations of the columns of A.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Full column rank r = n.

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

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Gram-Schmidt orthogonalization A = QR.

Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Length II x II.

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

• Linear combination cv + d w or L C jV j.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Reduced row echelon form R = rref(A).

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

• Singular matrix A.

A square matrix that has no inverse: det(A) = o.

• Special solutions to As = O.

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

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• 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.

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