 1.2.1: In Exercises 14, convert the given augmented matrix to theequivalen...
 1.2.2: In Exercises 14, convert the given augmented matrix to theequivalen...
 1.2.3: In Exercises 14, convert the given augmented matrix to theequivalen...
 1.2.4: In Exercises 14, convert the given augmented matrix to theequivalen...
 1.2.5: In Exercises 510, determine those matrices that are in echelonform,...
 1.2.6: In Exercises 510, determine those matrices that are in echelonform,...
 1.2.7: In Exercises 510, determine those matrices that are in echelonform,...
 1.2.8: In Exercises 510, determine those matrices that are in echelonform,...
 1.2.9: In Exercises 510, determine those matrices that are in echelonform,...
 1.2.10: In Exercises 510, determine those matrices that are in echelonform,...
 1.2.11: In Exercises 1114, the matrix on the right results after performing...
 1.2.12: In Exercises 1114, the matrix on the right results after performing...
 1.2.13: In Exercises 1114, the matrix on the right results after performing...
 1.2.14: In Exercises 1114, the matrix on the right results after performing...
 1.2.15: In Exercises 1518, a single row operation was performed on thematri...
 1.2.16: In Exercises 1518, a single row operation was performed on thematri...
 1.2.17: In Exercises 1518, a single row operation was performed on thematri...
 1.2.18: In Exercises 1518, a single row operation was performed on thematri...
 1.2.19: In Exercises 1926, convert the given system to an augmented matrixa...
 1.2.20: In Exercises 1926, convert the given system to an augmented matrixa...
 1.2.21: In Exercises 1926, convert the given system to an augmented matrixa...
 1.2.22: In Exercises 1926, convert the given system to an augmented matrixa...
 1.2.23: In Exercises 1926, convert the given system to an augmented matrixa...
 1.2.24: In Exercises 1926, convert the given system to an augmented matrixa...
 1.2.25: In Exercises 1926, convert the given system to an augmented matrixa...
 1.2.26: In Exercises 1926, convert the given system to an augmented matrixa...
 1.2.27: In Exercises 2730, convert the given system to an augmentedmatrix a...
 1.2.28: In Exercises 2730, convert the given system to an augmentedmatrix a...
 1.2.29: In Exercises 2730, convert the given system to an augmentedmatrix a...
 1.2.30: In Exercises 2730, convert the given system to an augmentedmatrix a...
 1.2.31: For each of Exercises 3136, suppose that the given row operationis ...
 1.2.32: For each of Exercises 3136, suppose that the given row operationis ...
 1.2.33: For each of Exercises 3136, suppose that the given row operationis ...
 1.2.34: For each of Exercises 3136, suppose that the given row operationis ...
 1.2.35: For each of Exercises 3136, suppose that the given row operationis ...
 1.2.36: For each of Exercises 3136, suppose that the given row operationis ...
 1.2.37: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 1.2.38: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 1.2.39: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 1.2.40: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 1.2.41: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 1.2.42: FIND AN EXAMPLE For Exercises 3742, find an example thatmeets the g...
 1.2.43: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 1.2.44: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 1.2.45: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 1.2.46: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 1.2.47: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 1.2.48: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 1.2.49: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 1.2.50: TRUE OR FALSE For Exercises 4350, determine if the statementis true...
 1.2.51: Suppose that the echelon form of an augmented matrix hasa pivot pos...
 1.2.52: Suppose that the echelon form of an augmented matrix hasa pivot pos...
 1.2.53: Show that if a linear system has two different solutions, thenit mu...
 1.2.54: Show that if a matrix has more rows than columns and is inechelon f...
 1.2.55: Show that a homogeneous linear system with more variablesthan equat...
 1.2.56: Show that each of the elementary operations on linear systems(see p...
 1.2.57: C In Exercises 5758 you are asked to find an interpolatingpolynomia...
 1.2.58: C In Exercises 5758 you are asked to find an interpolatingpolynomia...
 1.2.59: C Exercises 5960 refer to the cannonball scenario described atthe s...
 1.2.60: C Exercises 5960 refer to the cannonball scenario described atthe s...
 1.2.61: C In Exercises 6168, the given matrix is the augmented matrixfor a ...
 1.2.62: C In Exercises 6168, the given matrix is the augmented matrixfor a ...
 1.2.63: C In Exercises 6168, the given matrix is the augmented matrixfor a ...
 1.2.64: C In Exercises 6168, the given matrix is the augmented matrixfor a ...
 1.2.65: C In Exercises 6168, the given matrix is the augmented matrixfor a ...
 1.2.66: C In Exercises 6168, the given matrix is the augmented matrixfor a ...
 1.2.67: C In Exercises 6168, the given matrix is the augmented matrixfor a ...
 1.2.68: C In Exercises 6168, the given matrix is the augmented matrixfor a ...
Solutions for Chapter 1.2: Linear Systems and Matrices
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 1.2: Linear Systems and Matrices
Get Full SolutionsSince 68 problems in chapter 1.2: Linear Systems and Matrices have been answered, more than 16862 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. Chapter 1.2: Linear Systems and Matrices includes 68 full stepbystep solutions.

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

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.