 1.4.1: The volume of traffic for a collection of intersections is shownin ...
 1.4.2: C The volume of traffic for a collection of intersections isshown i...
 1.4.3: C The volume of traffic for a collection of intersections isshown i...
 1.4.4: C The volume of traffic for a collection of intersections isshown i...
 1.4.5: In Exercises 512, balance the given chemical equations.Hydrogen bur...
 1.4.6: In Exercises 512, balance the given chemical equations.Hydrogen and...
 1.4.7: In Exercises 512, balance the given chemical equations.Iron and oxy...
 1.4.8: In Exercises 512, balance the given chemical equations.Sodium and w...
 1.4.9: In Exercises 512, balance the given chemical equations.When propane...
 1.4.10: In Exercises 512, balance the given chemical equations.When acetyle...
 1.4.11: In Exercises 512, balance the given chemical equations.Potassium su...
 1.4.12: In Exercises 512, balance the given chemical equations.Manganese di...
 1.4.13: In Exercises 1316, find a model for planetary orbital period usingt...
 1.4.14: In Exercises 1316, find a model for planetary orbital period usingt...
 1.4.15: In Exercises 1316, find a model for planetary orbital period usingt...
 1.4.16: In Exercises 1316, find a model for planetary orbital period usingt...
 1.4.17: In Exercises 1718, the data given provides the distance requiredfor...
 1.4.18: In Exercises 1718, the data given provides the distance requiredfor...
 1.4.19: When using partial fractions to find antiderivatives in calculus, w...
 1.4.20: When using partial fractions to find antiderivatives in calculus, w...
 1.4.21: When using partial fractions to find antiderivatives in calculus, w...
 1.4.22: When using partial fractions to find antiderivatives in calculus, w...
 1.4.23: The points (1, 3) and (2, 6) lie on a line. Where does the linecros...
 1.4.24: The points (2, 1, 2), (1, 3, 12), and (4, 2, 3) lie on a uniqueplan...
 1.4.25: The equation for a parabola has the form y = ax2 + bx + c,where a, ...
 1.4.26: C Find a polynomial of the formf (x) = ax3 + bx2 + c x + dsuch that...
 1.4.27: C Find a polynomial of the formg (x) = ax4 + bx3 + c x2 + dx + esuc...
 1.4.28: C (Calculus required) In Exercises 2829, find the values of thecoef...
 1.4.29: C (Calculus required) In Exercises 2829, find the values of thecoef...
 1.4.30: C In Exercises 3031, a new LAI (for Linear Algebra Index)formula ha...
 1.4.31: C In Exercises 3031, a new LAI (for Linear Algebra Index)formula ha...
 1.4.32: . C The BCS ranking system was more complicated in 2001than in 2008...
Solutions for Chapter 1.4: Applications of Linear Systems
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 1.4: Applications of Linear Systems
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Chapter 1.4: Applications of Linear Systems includes 32 full stepbystep solutions. Since 32 problems in chapter 1.4: Applications of Linear Systems have been answered, more than 14535 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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!

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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

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

Factorization
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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

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·