 7.2.1: In Exercises 13 solve the lower triangular systems. X1 12X1  X2 =...
 7.2.2: In Exercises 13 solve the lower triangular systems. 2x1 = 4xl + X...
 7.2.3: In Exercises 13 solve the lower triangular systems. X1 = 23x1 + X...
 7.2.4: In Exercises 46 solve the upper triangular systems. X1 + X2 + X3 =...
 7.2.5: In Exercises 46 solve the upper triangular systems. 2X1  X2 + X3 ...
 7.2.6: In Exercises 46 solve the upper triangular systems. 3x1 + 2x2  x3...
 7.2.7: Let A = LU. The following sequences of transformations were used to...
 7.2.8: Let A = LU. Determine the row operations used to arrive at U for ea...
 7.2.9: In Exercises 916 solve the systems using the method of LU decompos...
 7.2.10: In Exercises 916 solve the systems using the method of LU decompos...
 7.2.11: In Exercises 916 solve the systems using the method of LU decompos...
 7.2.12: In Exercises 916 solve the systems using the method of LU decompos...
 7.2.13: In Exercises 916 solve the systems using the method of LU decompos...
 7.2.14: In Exercises 916 solve the systems using the method of LU decompos...
 7.2.15: In Exercises 916 solve the systems using the method of LU decompos...
 7.2.16: In Exercises 916 solve the systems using the method of LU decompos...
 7.2.17: In Exercises 1719 solve the systems using the method of LU decompo...
 7.2.18: In Exercises 1719 solve the systems using the method of LU decompo...
 7.2.19: In Exercises 1719 solve the systems using the method of LU decompo...
 7.2.20: In Exercises 20 and 21 solve the systems using the method of LU dec...
 7.2.21: In Exercises 20 and 21 solve the systems using the method of LU dec...
 7.2.22: Prove that the inverse (if it exists) of a lower triangular matrix ...
 7.2.23: Prove that the product of two lower triangular matrices is lower tr...
 7.2.24: Show that an LU decomposition of a matrix (if one exists) is not un...
 7.2.25: Consider the matrix E [ !] that defures the interchange of rows 2 a...
 7.2.26: Let AX= B be a system where A= LU andA is a 3 X 3 matrix. The formu...
Solutions for Chapter 7.2: The Method of LU Decomposition
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 7.2: The Method of LU Decomposition
Get Full SolutionsSince 26 problems in chapter 7.2: The Method of LU Decomposition have been answered, more than 8744 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Chapter 7.2: The Method of LU Decomposition includes 26 full stepbystep solutions.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Iterative method.
A sequence of steps intended to approach the desired solution.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).