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Solutions for Chapter 6.4: Matrix of a linear Transformation

Full solutions for Elementary Linear Algebra with Applications | 9th Edition

ISBN: 9780471669593

Solutions for Chapter 6.4: Matrix of a linear Transformation

Solutions for Chapter 6.4
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Textbook: Elementary Linear Algebra with Applications
Edition: 9
Author: Howard Anton, Chris Rorres
ISBN: 9780471669593

Chapter 6.4: Matrix of a linear Transformation includes 24 full step-by-step solutions. Since 24 problems in chapter 6.4: Matrix of a linear Transformation have been answered, more than 10224 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This expansive textbook survival guide covers the following chapters and their solutions.

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

    Parentheses can be removed to leave ABC.

  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • Cofactor Cij.

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

  • Conjugate Gradient Method.

    A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • Ellipse (or ellipsoid) x T Ax = 1.

    A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • Hypercube matrix pl.

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

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • 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.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • 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.

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

    Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

  • Row picture of Ax = b.

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

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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