 1.2.1: Matrix Size In Exercises 16, determine the size of the matrix.[1302...
 1.2.1.2.1: Matrix Size In Exercises 16, determine the size of the matrix.[1302...
 1.2.2: Matrix Size In Exercises 16, determine the size of the matrix.[2112]
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 1.2.3: Matrix Size In Exercises 16, determine the size of the matrix.[ 261...
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 1.2.4: Matrix Size In Exercises 16, determine the size of the matrix.[1]
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 1.2.5: Matrix Size In Exercises 16, determine the size of the matrix.[8211...
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 1.2.6: Matrix Size In Exercises 16, determine the size of the matrix.[1 2 ...
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 1.2.7: Elementary Row Operations In Exercises 710, identify the elementary...
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 1.2.11: Augmented Matrix In Exercises 1118, find the solution set of the sy...
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 1.2.19: RowEchelon Form In Exercises 1924, determine whether the matrix is...
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 1.2.1.2.25: System of Linear Equations In Exercises 2538, solve the system usin...
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 1.2.1.2.39: System of Linear Equations In Exercises 3942, use a software progra...
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 1.2.1.2.47: Finance A small software corporation borrowed $500,000 to expand it...
 1.2.47: Finance A small software corporation borrowed $500,000 to expand it...
 1.2.1.2.48: Tips A food server examines the amount of money earned in tips afte...
 1.2.48: Tips A food server examines the amount of money earned in tips afte...
 1.2.1.2.49: Matrix Representation In Exercises 49 and 50, assume that the matri...
 1.2.49: Matrix Representation In Exercises 49 and 50, assume that the matri...
 1.2.1.2.50: Matrix Representation In Exercises 49 and 50, assume that the matri...
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 1.2.1.2.51: Coefficient Design In Exercises 51 and 52, find values of a, b, and...
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 1.2.1.2.53: The system below has one solution: x = 1, y = 1,and z = 2.4x x +x 2...
 1.2.53: The system below has one solution: x = 1, y = 1,and z = 2.4x x +x 2...
 1.2.1.2.54: Assume the system below has a unique solution.a11x1 +a21x1 +a31x1 +...
 1.2.54: Assume the system below has a unique solution.a11x1 +a21x1 +a31x1 +...
 1.2.1.2.55: Row Equivalence In Exercises 55 and 56, find the reduced rowechelo...
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 1.2.1.2.56: Row Equivalence In Exercises 55 and 56, find the reduced rowechelo...
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 1.2.1.2.57: Writing Describe all possible 2 2 reduced rowechelon matrices. Sup...
 1.2.57: Writing Describe all possible 2 2 reduced rowechelon matrices. Sup...
 1.2.1.2.58: Writing Describe all possible 3 3 reduced rowechelon matrices. Sup...
 1.2.58: Writing Describe all possible 3 3 reduced rowechelon matrices. Sup...
 1.2.1.2.59: True or False? In Exercises 59 and 60, determine whether each state...
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 1.2.60: True or False? In Exercises 59 and 60, determine whether each state...
 1.2.1.2.61: Writing Is it possible for a system of linear equations with fewer ...
 1.2.61: Writing Is it possible for a system of linear equations with fewer ...
 1.2.1.2.62: Writing Does a matrix have a unique rowechelon form? Illustrate yo...
 1.2.62: Writing Does a matrix have a unique rowechelon form? Illustrate yo...
 1.2.1.2.63: Row Equivalence In Exercises 63 and 64, determineconditions on a, b...
 1.2.63: Row Equivalence In Exercises 63 and 64, determineconditions on a, b...
 1.2.1.2.64: Row Equivalence In Exercises 63 and 64, determineconditions on a, b...
 1.2.64: Row Equivalence In Exercises 63 and 64, determineconditions on a, b...
 1.2.1.2.65: Homogeneous System In Exercises 65 and 66, find all values of (the ...
 1.2.65: Homogeneous System In Exercises 65 and 66, find all values of (the ...
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 1.2.1.2.67: The augmented matrix represents a system of linearequations that ha...
 1.2.67: The augmented matrix represents a system of linearequations that ha...
 1.2.1.2.68: CAPSTONE In your own words, describe the difference between a matri...
 1.2.68: CAPSTONE In your own words, describe the difference between a matri...
 1.2.1.2.69: . Writing Consider the 2 2 matrix [acbd].Perform the sequence of ro...
 1.2.69: . Writing Consider the 2 2 matrix [acbd].Perform the sequence of ro...
 1.2.1.2.70: Writing Describe the rowechelon form of an augmented matrix that c...
 1.2.70: Writing Describe the rowechelon form of an augmented matrix that c...
Solutions for Chapter 1.2: Gaussian Elimination and GaussJordan Elimination
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 1.2: Gaussian Elimination and GaussJordan Elimination
Get Full SolutionsChapter 1.2: Gaussian Elimination and GaussJordan Elimination includes 140 full stepbystep solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. This expansive textbook survival guide covers the following chapters and their solutions. Since 140 problems in chapter 1.2: Gaussian Elimination and GaussJordan Elimination have been answered, more than 44401 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8.

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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

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·

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.