 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...
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 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 A=[: !] and x=['::lShow that the linear system Ax = 0 has only ...
 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
 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: Exerr:;st's 24 lI/u/ 25 (1ft 0flliolla/.(8) FOnllUlate the de finit...
 2.2.25: Exerr:;st's 24 lI/u/ 25 (1ft 0flliolla/.Prove that every III >( /I ...
 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: (CIt/CII/IIS Reqllired) Construct 11 linear system of equations to ...
 2.2.32: (Ca(ClI(lIs Reqllired) Co nstruct a linear system o f o!q uations t...
 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.(I ...
 2.2.41: III Exell"ise.l 40 alld 41 .. w/l'e each gil't'1! linear system.x +...
 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 Lint!ar Systt!ms
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 2.2: Solving Lint!ar Systt!ms
Get Full SolutionsElementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. Chapter 2.2: Solving Lint!ar Systt!ms includes 46 full stepbystep solutions. Since 46 problems in chapter 2.2: Solving Lint!ar Systt!ms have been answered, more than 13568 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: Elementary Linear Algebra with Applications, edition: 9.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

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.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

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.

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