- 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 one-way streets with tra...
- 1.9.4: The accompanying figure shows a network of one-way 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
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
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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, off-diagonal entries = 0.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
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 = Q-1 = 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. A-I 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.