 1.1: In Exercises 18, determine whether the equation is linear in the va...
 1.2: In Exercises 18, determine whether the equation is linear in the va...
 1.3: In Exercises 18, determine whether the equation is linear in the va...
 1.4: In Exercises 18, determine whether the equation is linear in the va...
 1.5: In Exercises 18, determine whether the equation is linear in the va...
 1.6: In Exercises 18, determine whether the equation is linear in the va...
 1.7: In Exercises 18, determine whether the equation is linear in the va...
 1.8: In Exercises 18, determine whether the equation is linear in the va...
 1.9: In Exercises 9 and 10, find a parametric representation of the solu...
 1.10: In Exercises 9 and 10, find a parametric representation of the solu...
 1.11: In Exercises 1122, solve the system of linear equations.
 1.12: In Exercises 1122, solve the system of linear equations.
 1.13: In Exercises 1122, solve the system of linear equations.
 1.14: In Exercises 1122, solve the system of linear equations.
 1.15: In Exercises 1122, solve the system of linear equations.
 1.16: In Exercises 1122, solve the system of linear equations.
 1.17: In Exercises 1122, solve the system of linear equations.
 1.18: In Exercises 1122, solve the system of linear equations.
 1.19: In Exercises 1122, solve the system of linear equations.
 1.20: In Exercises 1122, solve the system of linear equations.
 1.21: In Exercises 1122, solve the system of linear equations.
 1.22: In Exercises 1122, solve the system of linear equations.
 1.23: In Exercises 23 and 24, determine the size of the matrix.
 1.24: In Exercises 23 and 24, determine the size of the matrix.
 1.25: In Exercises 2528, determine whether the matrix is in rowechelon f...
 1.26: In Exercises 2528, determine whether the matrix is in rowechelon f...
 1.27: In Exercises 2528, determine whether the matrix is in rowechelon f...
 1.28: In Exercises 2528, determine whether the matrix is in rowechelon f...
 1.29: In Exercises 29 and 30, find the solution set of the system of line...
 1.30: In Exercises 29 and 30, find the solution set of the system of line...
 1.31: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.32: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.33: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.34: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.35: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.36: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.37: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.38: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.39: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.40: In Exercises 3140, solve the system using either Gaussian eliminati...
 1.41: In Exercises 41 46, use the matrix capabilities of a graphing utili...
 1.42: In Exercises 41 46, use the matrix capabilities of a graphing utili...
 1.43: In Exercises 41 46, use the matrix capabilities of a graphing utili...
 1.44: In Exercises 41 46, use the matrix capabilities of a graphing utili...
 1.45: In Exercises 41 46, use the matrix capabilities of a graphing utili...
 1.46: In Exercises 41 46, use the matrix capabilities of a graphing utili...
 1.47: In Exercises 4750, solve the homogeneous system of linear equations.
 1.48: In Exercises 4750, solve the homogeneous system of linear equations.
 1.49: In Exercises 4750, solve the homogeneous system of linear equations.
 1.50: In Exercises 4750, solve the homogeneous system of linear equations.
 1.51: Determine the value of k such that the system of linear equations i...
 1.52: Determine the value of k such that the system of linear equations h...
 1.53: Find conditions on a and b such that the system of linear equations...
 1.54: Find (if possible) conditions on a, b, and c such that the system o...
 1.55: Writing Describe a method for showing that two matrices are rowequ...
 1.56: Writing Describe all possible reduced rowechelon matrices. Support...
 1.57: Let Find the reduced rowechelon form of the matrix.
 1.58: Find all values of for which the homogeneous system of linear equat...
 1.59: (a) The solution set of a linear equation can be parametrically rep...
 1.60: (a) A homogeneous system of linear equations must have at least one...
 1.61: The University of Tennessee Lady Volunteers defeated the Rutgers Un...
 1.62: In Super Bowl I, on January 15, 1967, the Green Bay Packers defeate...
 1.63: In Exercises 63 and 64, use a system of equations to write the part...
 1.64: In Exercises 63 and 64, use a system of equations to write the part...
 1.65: In Exercises 65 and 66, (a) determine the polynomial whose graph pa...
 1.66: In Exercises 65 and 66, (a) determine the polynomial whose graph pa...
 1.67: A company has sales (measured in millions) of $50, $60, and $75 dur...
 1.68: The polynomial function is zerowhen and 4. What are the values of and
 1.69: A wildlife management team studied the population of deer in one sm...
 1.70: A research team studied the average monthly temperatures of a small...
 1.71: Determine the currents and for the electrical network shown in Figu...
 1.72: The flow through a network is shown in Figure 1.24. (a) Solve the s...
Solutions for Chapter 1: Systems of Linear Equations
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 1: Systems of Linear Equations
Get Full SolutionsSince 72 problems in chapter 1: Systems of Linear Equations have been answered, more than 18456 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6. Chapter 1: Systems of Linear Equations includes 72 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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).

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·