 2.2.1: Each of the given linear systems is in row echelon form. Solve the ...
 2.2.2: Each of the given linear systems is in row echelon form. Solve the ...
 2.2.3: Each of the given linear systems is in reJuceJ row eche Ion form. S...
 2.2.4: Each of the given linear systems is in reJuced row eche Ion form. S...
 2.2.5: ConsiJer the linear system x + y +2z=  1 x  2)' + z = 5 3x+ y + ...
 2.2.6: Repeat Exercise 5 for each of the following linear systems: (a) x +...
 2.2.7: In :ierciJeJ 7 IhIV1I8h 9, mll'e tiIe linear Jptl'm. lI'ilh Ihe gir...
 2.2.8: In :ierciJeJ 7 IhIV1I8h 9, mll'e tiIe linear Jptl'm. lI'ilh Ihe gir...
 2.2.9: In :ierciJeJ 7 IhIV1I8h 9, mll'e tiIe linear Jptl'm. lI'ilh Ihe gir...
 2.2.10: Find a 2 x I matrix x with entries not all zero such th
 2.2.11: Find a 2 x I matrix x with entries not all zero such that Ax = 3x. ...
 2.2.12: Find a 3 x I matrix x with entries not all zero such that
 2.2.13: Find a 3 x I matrix x with entries not all zero such that
 2.2.14: In the following linear system. determine all values of a for which...
 2.2.15: Repeat Exercise 14 for the linear system { + y + 2x + 3y + :=2 2z ...
 2.2.16: Repeat Exercise 14 for the linear system x+ y+ x + 2)' + z =2 z =3 ...
 2.2.17: Repeat Exercise 14 for the linear system x + y = 3 x+(a2  8) y =a.
 2.2.18: Let 19. A=[: !] and x=['::l Show that the linear system Ax = 0 has ...
 2.2.19: Show that A = e d is row equivalent to 11 if and only if ad  be f O.
 2.2.20: Let I : RJ + RJ be the matrix transfomlation defined 21. 22. by
 2.2.21: Lei f. ~ ..... R' be Il,e lIlal' ;). IJansfo"nat;oll defined by [~]...
 2.2.22: Let I : RJ + RJ be the matrix transformation defined by , 2 Find ...
 2.2.23: Let /: RJ + RJ be the matrix transformation defined by ! ) ~ ~ ;]...
 2.2.24: (8) FOnllUlate the de finiti ons of column echelon form and reduced...
 2.2.25: Prove that every III >( /I matrix is column equivalent to a unique ...
 2.2.26: Find an equation relating (I. b. and e so that the linear system x ...
 2.2.27: Find an equation relating tl , b, and e so that the linear system 2...
 2.2.28: Sho ..... that the homogeneous syMem (tl  r).f+ d),= O u+(b  r )y...
 2.2.29: Let Ax = b. b =F o. be a consistent linear system. (8) Show that if...
 2.2.30: Determine the quadr . tic interpolant to each of the given d"ta set...
 2.2.31: Construct 11 linear system of equations to detcnlline a ~adrntic po...
 2.2.32: Co nstruct a linear system o f o!q uations to dc tenlline a qJadrat...
 2.2.33: Determine the temperatures at the interior points Ti i = l. 2. 3. 4...
 2.2.34: Determine the planar location (x. y) of a GPS receiver. using coord...
 2.2.35: The location of a GPS receiver in a twodimensional system is (4. ...
 2.2.36: The location of a GPS receiver in a twodimensional system is (6. 8...
 2.2.37: Suppose you have a "special edition" GPS receiver for rwn_rl;m<". s...
 2.2.38: Rust is formed when there is a chemical reaction between iron and o...
 2.2.39: Ethane is a gas similar to methane that bums in oxygen to give carb...
 2.2.40: III Exell"ise.l 40 alld 41 .. w/l'e each gil't'1! linear system.
 2.2.41: III Exell"ise.l 40 alld 41 .. w/l'e each gil't'1! linear system.
 2.2.42: III Exercises 42 alld 43. soll'e each lif!tlllr .I'ystem whose augm...
 2.2.43: III Exercises 42 alld 43. soll'e each lif!tlllr .I'ystem whose augm...
 2.2.44: Determine whether the software you are using has acommand for compu...
 2.2.45: Detennine whether the software you are using has a command for comp...
 2.2.46: Determine whether the software you are using has a graphing option ...
Solutions for Chapter 2.2: Solving Linear Systems
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 2.2: Solving Linear Systems
Get Full SolutionsElementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 46 problems in chapter 2.2: Solving Linear Systems have been answered, more than 10174 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.2: Solving Linear Systems includes 46 full stepbystep solutions.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

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 •

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.