 8.8.1: In Exercises 1 and 2, fill in the blank. 1. _______ is a method for...
 8.8.2: In Exercises 1 and 2, fill in the blank. 2. A message written accor...
 8.8.3: In Exercises 3 and 4, consider three points (x1, y1), (x2, y2), and...
 8.8.4: In Exercises 3 and 4, consider three points (x1, y1), (x2, y2), and...
 8.8.5: In Exercises 510, use a determinant to find the area of the figure ...
 8.8.6: In Exercises 510, use a determinant to find the area of the figure ...
 8.8.7: In Exercises 510, use a determinant to find the area of the figure ...
 8.8.8: In Exercises 510, use a determinant to find the area of the figure ...
 8.8.9: In Exercises 510, use a determinant to find the area of the figure ...
 8.8.10: In Exercises 510, use a determinant to find the area of the figure ...
 8.8.11: In Exercises 11 and 12, find x or y such that the triangle has an a...
 8.8.12: In Exercises 11 and 12, find x or y such that the triangle has an a...
 8.8.13: In Exercises 1316, use a determinant to determine whether the point...
 8.8.14: In Exercises 1316, use a determinant to determine whether the point...
 8.8.15: In Exercises 1316, use a determinant to determine whether the point...
 8.8.16: In Exercises 1316, use a determinant to determine whether the point...
 8.8.17: In Exercises 17 and 18, find x or y such that the points are collin...
 8.8.18: In Exercises 17 and 18, find x or y such that the points are collin...
 8.8.19: In Exercises 1924, use Cramers Rule to solve (if possible) the syst...
 8.8.20: In Exercises 1924, use Cramers Rule to solve (if possible) the syst...
 8.8.21: In Exercises 1924, use Cramers Rule to solve (if possible) the syst...
 8.8.22: In Exercises 1924, use Cramers Rule to solve (if possible) the syst...
 8.8.23: In Exercises 1924, use Cramers Rule to solve (if possible) the syst...
 8.8.24: In Exercises 1924, use Cramers Rule to solve (if possible) the syst...
 8.8.25: In Exercises 25 and 26, solve the system of equations using (a) Gau...
 8.8.26: In Exercises 25 and 26, solve the system of equations using (a) Gau...
 8.8.27: The retail sales of family clothing stores in the United States fro...
 8.8.28: The retail sales y (in billions of dollars) of stores selling auto ...
 8.8.29: In Exercises 29 and 30, (a) write the uncoded 1 2 row matrices for ...
 8.8.30: In Exercises 29 and 30, (a) write the uncoded 1 2 row matrices for ...
 8.8.31: In Exercises 31 and 32, (a) write the uncoded 1 3 row matrices for ...
 8.8.32: In Exercises 31 and 32, (a) write the uncoded 1 3 row matrices for ...
 8.8.33: In Exercises 33 and 34, use A1 to decode the cryptogram. 33. A = [ ...
 8.8.34: In Exercises 33 and 34, use A1 to decode the cryptogram. 34. A = [ ...
 8.8.35: The following cryptogram was encoded with a 2 2 matrix. 5 2 25 11 2...
 8.8.36: The following cryptogram was encoded with a 2 2 matrix. 5 2 25 11 2...
 8.8.37: In Exercises 37 and 38, determine whether the statement is true or ...
 8.8.38: In Exercises 37 and 38, determine whether the statement is true or ...
 8.8.39: Describe a way to use an invertible n n matrix to encode a message ...
 8.8.40: At this point in the text, you have learned several methods for fin...
 8.8.41: In Exercises 4144, find the general form of the equation of the lin...
 8.8.42: In Exercises 4144, find the general form of the equation of the lin...
 8.8.43: In Exercises 4144, find the general form of the equation of the lin...
 8.8.44: In Exercises 4144, find the general form of the equation of the lin...
Solutions for Chapter 8.8: Linear Systems and Matrices
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 8.8: Linear Systems and Matrices
Get Full SolutionsSince 44 problems in chapter 8.8: Linear Systems and Matrices have been answered, more than 64427 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Chapter 8.8: Linear Systems and Matrices includes 44 full stepbystep solutions.

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.

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!

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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.

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).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.