 1.1.1.1.1: In Exercises 16, determine whether the equation is linear in the va...
 1.1.1: In Exercises 16, determine whether the equation is linear in the va...
 1.1.1.1.2: In Exercises 16, determine whether the equation is linear in the va...
 1.1.2: In Exercises 16, determine whether the equation is linear in the va...
 1.1.1.1.3: In Exercises 16, determine whether the equation is linear in the va...
 1.1.3: In Exercises 16, determine whether the equation is linear in the va...
 1.1.1.1.4: In Exercises 16, determine whether the equation is linear in the va...
 1.1.4: In Exercises 16, determine whether the equation is linear in the va...
 1.1.1.1.5: In Exercises 16, determine whether the equation is linear in the va...
 1.1.5: In Exercises 16, determine whether the equation is linear in the va...
 1.1.1.1.6: In Exercises 16, determine whether the equation is linear in the va...
 1.1.6: In Exercises 16, determine whether the equation is linear in the va...
 1.1.1.1.7: In Exercises 710, find a parametric representation of the solution ...
 1.1.7: In Exercises 710, find a parametric representation of the solution ...
 1.1.1.1.8: In Exercises 710, find a parametric representation of the solution ...
 1.1.8: In Exercises 710, find a parametric representation of the solution ...
 1.1.1.1.9: In Exercises 710, find a parametric representation of the solution ...
 1.1.9: In Exercises 710, find a parametric representation of the solution ...
 1.1.1.1.10: In Exercises 710, find a parametric representation of the solution ...
 1.1.10: In Exercises 710, find a parametric representation of the solution ...
 1.1.1.1.11: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.11: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.12: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.12: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.13: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.13: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.14: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.14: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.15: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.15: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.16: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.16: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.17: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.17: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.18: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.18: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.19: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.19: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.20: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.20: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.21: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.21: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.22: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.22: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.23: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.23: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.24: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.1.1.24: In Exercises 1124, graph the system of linear equations. Solve the ...
 1.1.25: In Exercises 2530, use backsubstitution to solve the system.x1 x2 =...
 1.1.1.1.25: In Exercises 2530, use backsubstitution to solve the system.x1 x2 =...
 1.1.26: In Exercises 2530, use backsubstitution to solve the system.2x1 4x2...
 1.1.1.1.26: In Exercises 2530, use backsubstitution to solve the system.2x1 4x2...
 1.1.27: In Exercises 2530, use backsubstitution to solve the system.x + y 2...
 1.1.1.1.27: In Exercises 2530, use backsubstitution to solve the system.x + y 2...
 1.1.28: In Exercises 2530, use backsubstitution to solve the system.x y3y +...
 1.1.1.1.28: In Exercises 2530, use backsubstitution to solve the system.x y3y +...
 1.1.29: In Exercises 2530, use backsubstitution to solve the system.5x1 +2x...
 1.1.1.1.29: In Exercises 2530, use backsubstitution to solve the system.5x1 +2x...
 1.1.30: In Exercises 2530, use backsubstitution to solve the system.x1 + x2...
 1.1.1.1.30: In Exercises 2530, use backsubstitution to solve the system.x1 + x2...
 1.1.31: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.1.1.31: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.32: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.1.1.32: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.33: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.1.1.33: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.34: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.1.1.34: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.35: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.1.1.35: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.36: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.1.1.36: s In Exercises 3136, complete parts(a)(e) for the system of equatio...
 1.1.37: In Exercises 3756, solve the system of linear equations.x1 3x1 x2 =...
 1.1.1.1.37: In Exercises 3756, solve the system of linear equations.x1 3x1 x2 =...
 1.1.38: In Exercises 3756, solve the system of linear equations.3x + 2y =6x...
 1.1.1.1.38: In Exercises 3756, solve the system of linear equations.3x + 2y =6x...
 1.1.39: In Exercises 3756, solve the system of linear equations.3u +u +v = ...
 1.1.1.1.39: In Exercises 3756, solve the system of linear equations.3u +u +v = ...
 1.1.40: In Exercises 3756, solve the system of linear equations.x1 2x2 = 06...
 1.1.1.1.40: In Exercises 3756, solve the system of linear equations.x1 2x2 = 06...
 1.1.41: In Exercises 3756, solve the system of linear equations.9x 3y = 115...
 1.1.1.1.41: In Exercises 3756, solve the system of linear equations.9x 3y = 115...
 1.1.42: In Exercises 3756, solve the system of linear equations.23x1 +4x1 +...
 1.1.1.1.42: In Exercises 3756, solve the system of linear equations.23x1 +4x1 +...
 1.1.43: In Exercises 3756, solve the system of linear equations.x 24+ y 13 ...
 1.1.1.1.43: In Exercises 3756, solve the system of linear equations.x 24+ y 13 ...
 1.1.44: In Exercises 3756, solve the system of linear equations.x1 + 43+x2 ...
 1.1.1.1.44: In Exercises 3756, solve the system of linear equations.x1 + 43+x2 ...
 1.1.45: In Exercises 3756, solve the system of linear equations.. 0.02x1 0....
 1.1.1.1.45: In Exercises 3756, solve the system of linear equations.. 0.02x1 0....
 1.1.46: In Exercises 3756, solve the system of linear equations.0.05x1 0.03...
 1.1.1.1.46: In Exercises 3756, solve the system of linear equations.0.05x1 0.03...
 1.1.47: In Exercises 3756, solve the system of linear equations.xx2x+y2y z ...
 1.1.1.1.47: In Exercises 3756, solve the system of linear equations.xx2x+y2y z ...
 1.1.48: In Exercises 3756, solve the system of linear equations.x +x +4x +y...
 1.1.1.1.48: In Exercises 3756, solve the system of linear equations.x +x +4x +y...
 1.1.49: In Exercises 3756, solve the system of linear equations.3x1 x1 +2x1...
 1.1.1.1.49: In Exercises 3756, solve the system of linear equations.3x1 x1 +2x1...
 1.1.50: In Exercises 3756, solve the system of linear equations.5x1 2x1 +x1...
 1.1.1.1.50: In Exercises 3756, solve the system of linear equations.5x1 2x1 +x1...
 1.1.51: In Exercises 3756, solve the system of linear equations.2x14x12x1++...
 1.1.1.1.51: In Exercises 3756, solve the system of linear equations.2x14x12x1++...
 1.1.52: In Exercises 3756, solve the system of linear equations.. x14x12x1 ...
 1.1.1.1.52: In Exercises 3756, solve the system of linear equations.. x14x12x1 ...
 1.1.53: In Exercises 3756, solve the system of linear equations.x 5x 3y +15...
 1.1.1.1.53: In Exercises 3756, solve the system of linear equations.x 5x 3y +15...
 1.1.54: In Exercises 3756, solve the system of linear equations.x1 2x2 +3x1...
 1.1.1.1.54: In Exercises 3756, solve the system of linear equations.x1 2x2 +3x1...
 1.1.55: In Exercises 3756, solve the system of linear equations.x +2x +3x +...
 1.1.1.1.55: In Exercises 3756, solve the system of linear equations.x +2x +3x +...
 1.1.56: In Exercises 3756, solve the system of linear equations.x13x1 4x2x2...
 1.1.1.1.56: In Exercises 3756, solve the system of linear equations.x13x1 4x2x2...
 1.1.57: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.1.1.57: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.58: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.1.1.58: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.59: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.1.1.59: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.60: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.1.1.60: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.61: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.1.1.61: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.62: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.1.1.62: In Exercises 5762, use a software program or a graphing utility to ...
 1.1.63: In Exercises 6366, state why the system of equations must have at l...
 1.1.1.1.63: In Exercises 6366, state why the system of equations must have at l...
 1.1.64: In Exercises 6366, state why the system of equations must have at l...
 1.1.1.1.64: In Exercises 6366, state why the system of equations must have at l...
 1.1.65: In Exercises 6366, state why the system of equations must have at l...
 1.1.1.1.65: In Exercises 6366, state why the system of equations must have at l...
 1.1.66: In Exercises 6366, state why the system of equations must have at l...
 1.1.1.1.66: In Exercises 6366, state why the system of equations must have at l...
 1.1.67: Nutrition One eightounce glass of apple juice andone eightounce g...
 1.1.1.1.67: Nutrition One eightounce glass of apple juice andone eightounce g...
 1.1.68: Airplane Speed Two planes start from Los AngelesInternational Airpo...
 1.1.1.1.68: Airplane Speed Two planes start from Los AngelesInternational Airpo...
 1.1.69: True or False? In Exercises 69 and 70, determine whether each state...
 1.1.1.1.69: True or False? In Exercises 69 and 70, determine whether each state...
 1.1.70: True or False? In Exercises 69 and 70, determine whether each state...
 1.1.1.1.70: True or False? In Exercises 69 and 70, determine whether each state...
 1.1.71: Find a system of two equations in two variables, x1 andx2, that has...
 1.1.1.1.71: Find a system of two equations in two variables, x1 andx2, that has...
 1.1.72: Find a system of two equations in three variables,x1, x2, and x3, t...
 1.1.1.1.72: Find a system of two equations in three variables,x1, x2, and x3, t...
 1.1.73: Substitution In Exercises 7376, solve the system of equations by fi...
 1.1.1.1.73: Substitution In Exercises 7376, solve the system of equations by fi...
 1.1.74: Substitution In Exercises 7376, solve the system of equations by fi...
 1.1.1.1.74: Substitution In Exercises 7376, solve the system of equations by fi...
 1.1.75: Substitution In Exercises 7376, solve the system of equations by fi...
 1.1.1.1.75: Substitution In Exercises 7376, solve the system of equations by fi...
 1.1.76: Substitution In Exercises 7376, solve the system of equations by fi...
 1.1.1.1.76: Substitution In Exercises 7376, solve the system of equations by fi...
 1.1.77: Trigonometric Coefficients In Exercises 77 and 78, solve the system...
 1.1.1.1.77: Trigonometric Coefficients In Exercises 77 and 78, solve the system...
 1.1.78: Trigonometric Coefficients In Exercises 77 and 78, solve the system...
 1.1.1.1.78: Trigonometric Coefficients In Exercises 77 and 78, solve the system...
 1.1.79: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.1.1.79: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.80: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.1.1.80: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.81: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.1.1.81: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.82: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.1.1.82: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.83: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.1.1.83: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.84: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.1.1.84: Coefficient Design In Exercises 7984, determine the value(s) of k s...
 1.1.85: Determine the values of k such that the system of linearequations d...
 1.1.1.1.85: Determine the values of k such that the system of linearequations d...
 1.1.86: CAPSTONE Find values of a, b, and c suchthat the system of linear e...
 1.1.1.1.86: CAPSTONE Find values of a, b, and c suchthat the system of linear e...
 1.1.87: Writing Consider the system of linear equations in xand y.a1x + b1y...
 1.1.1.1.87: Writing Consider the system of linear equations in xand y.a1x + b1y...
 1.1.88: Writing Explain why the system of linear equations in Exercise 87 m...
 1.1.1.1.88: Writing Explain why the system of linear equations in Exercise 87 m...
 1.1.89: Show that if ax2 + bx + c = 0 for all x, then a = b = c = 0.
 1.1.1.1.89: Show that if ax2 + bx + c = 0 for all x, then a = b = c = 0.
 1.1.90: Consider the system of linear equations in x and y.ax +cx +by =dy =...
 1.1.1.1.90: Consider the system of linear equations in x and y.ax +cx +by =dy =...
 1.1.91: Discovery In Exercises 91 and 92, sketch the lines represented by t...
 1.1.1.1.91: Discovery In Exercises 91 and 92, sketch the lines represented by t...
 1.1.92: Discovery In Exercises 91 and 92, sketch the lines represented by t...
 1.1.1.1.92: Discovery In Exercises 91 and 92, sketch the lines represented by t...
 1.1.93: Writing In Exercises 93 and 94, the graphs of the two equations app...
 1.1.1.1.93: Writing In Exercises 93 and 94, the graphs of the two equations app...
 1.1.94: Writing In Exercises 93 and 94, the graphs of the two equations app...
 1.1.1.1.94: Writing In Exercises 93 and 94, the graphs of the two equations app...
Solutions for Chapter 1.1: Introduction to Systems of Linear Equations
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 1.1: Introduction to Systems of Linear Equations
Get Full SolutionsElementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. Chapter 1.1: Introduction to Systems of Linear Equations includes 188 full stepbystep solutions. Since 188 problems in chapter 1.1: Introduction to Systems of Linear Equations have been answered, more than 42349 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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

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

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).

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!

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

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

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