×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide

Solutions for Chapter 1.3: Applications of Systems of Linear Equations

Elementary Linear Algebra | 8th Edition | ISBN: 9781305658004 | Authors: Ron Larson

Full solutions for Elementary Linear Algebra | 8th Edition

ISBN: 9781305658004

Elementary Linear Algebra | 8th Edition | ISBN: 9781305658004 | Authors: Ron Larson

Solutions for Chapter 1.3: Applications of Systems of Linear Equations

Solutions for Chapter 1.3
4 5 0 264 Reviews
25
0
Textbook: Elementary Linear Algebra
Edition: 8
Author: Ron Larson
ISBN: 9781305658004

This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. Since 84 problems in chapter 1.3: Applications of Systems of Linear Equations have been answered, more than 42732 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.3: Applications of Systems of Linear Equations includes 84 full step-by-step solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • 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!

  • Cofactor Cij.

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

  • 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

  • Diagonalizable matrix A.

    Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I 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).

  • Hankel matrix H.

    Constant along each antidiagonal; hij depends on i + j.

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • Kronecker product (tensor product) A ® B.

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

  • 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.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • 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.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Spanning set.

    Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

  • Vector addition.

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

×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
×
Reset your password