 1.9.1: The accompanying figure shows a network in which the flow rate and ...
 1.9.2: The accompanying figure shows known flow rates of hydrocarbons into...
 1.9.3: The accompanying figure shows a network of oneway streets with tra...
 1.9.4: The accompanying figure shows a network of oneway streets with tra...
 1.9.5: In Exercises 58, analyze the given electrical circuits by finding t...
 1.9.6: In Exercises 58, analyze the given electrical circuits by finding t...
 1.9.7: In Exercises 58, analyze the given electrical circuits by finding t...
 1.9.8: In Exercises 58, analyze the given electrical circuits by finding t...
 1.9.9: In Exercises 912, write a balanced equation for the given chemical ...
 1.9.10: In Exercises 912, write a balanced equation for the given chemical ...
 1.9.11: In Exercises 912, write a balanced equation for the given chemical ...
 1.9.12: In Exercises 912, write a balanced equation for the given chemical ...
 1.9.13: Find the quadratic polynomial whose graph passes through the points...
 1.9.14: Find the quadratic polynomial whose graph passes through the points...
 1.9.15: Find the cubic polynomial whose graph passes through the points (1,...
 1.9.16: The accompanying figure shows the graph of a cubic polynomial. Find...
 1.9.17: (a) Find an equation that represents the family of all seconddegree...
 1.9.18: In this section we have selected only a few applications of linear ...
Solutions for Chapter 1.9: Applications of Linear Systems
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 1.9: Applications of Linear Systems
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Since 18 problems in chapter 1.9: Applications of Linear Systems have been answered, more than 16782 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. Chapter 1.9: Applications of Linear Systems includes 18 full stepbystep solutions.

Column space C (A) =
space of all combinations of the columns 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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Elimination matrix = Elementary matrix Eij.
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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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)·(b  Ax) = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.