×
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
×

# Solutions for Chapter 20.1: Linear Systems: Gauss EliminatIOn

## Full solutions for Advanced Engineering Mathematics | 9th Edition

ISBN: 9780471488859

Solutions for Chapter 20.1: Linear Systems: Gauss EliminatIOn

Solutions for Chapter 20.1
4 5 0 238 Reviews
24
4
##### ISBN: 9780471488859

Since 16 problems in chapter 20.1: Linear Systems: Gauss EliminatIOn have been answered, more than 46116 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 20.1: Linear Systems: Gauss EliminatIOn includes 16 full step-by-step solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859.

Key Math Terms and definitions covered in this textbook
• Adjacency matrix of a graph.

Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

• 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.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of 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.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Linear combination cv + d w or L C jV j.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• Matrix multiplication AB.

The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

• Orthogonal matrix Q.

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

• 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.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• Trace of A

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

• 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.

×