 7.2.1: Let A = 1 1 1 2 4 1 3 1 2 Factor A into a product LU, where L is lo...
 7.2.2: Let A be the matrix in Exercise 1. Use the LU factorization of A to...
 7.2.3: Let A and B be n n matrices and let x Rn. (a) How many scalar addit...
 7.2.4: Let A Rmn, B Rnr , and x, y Rn. Suppose that the product AxyTB is c...
 7.2.5: Let Eki be the elementary matrix formed by subtracting times the it...
 7.2.6: Let A be an n n matrix with triangular factorization LU. Show that ...
 7.2.7: If A is a symmetric nn matrix with triangular factorization LU, the...
 7.2.8: Write an algorithm for solving the tridiagonal system a1 b1 c1 a2 ....
 7.2.9: Let A = LU, where L is lower triangular with 1s on the diagonal and...
 7.2.10: Suppose that A1 and the LU factorization of A have already been det...
 7.2.11: Let A be a 3 3 matrix, and assume that A can be transformed into a ...
Solutions for Chapter 7.2: Gaussian Elimination
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 7.2: Gaussian Elimination
Get Full SolutionsChapter 7.2: Gaussian Elimination includes 11 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Since 11 problems in chapter 7.2: Gaussian Elimination have been answered, more than 4937 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This expansive textbook survival guide covers the following chapters and their solutions.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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