 7.2.1: Let A = 111 241 3 1 2 Factor A into a product LU, where L is lower ...
 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 add...
 7.2.4: Let A Rmn, B Rnr , and x, y Rn. Suppose that the product AxyT B is ...
 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 l...
Solutions for Chapter 7.2: Gaussian Elimination
Full solutions for Linear Algebra with Applications  9th Edition
ISBN: 9780321962218
Solutions for Chapter 7.2: Gaussian Elimination
Get Full SolutionsSince 11 problems in chapter 7.2: Gaussian Elimination have been answered, more than 11948 students have viewed full stepbystep solutions from this chapter. Chapter 7.2: Gaussian Elimination includes 11 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. 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: 9780321962218.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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 IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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.

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

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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