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Solutions for Chapter 6.3: The Kernel and Range of a Linear Transformation

Full solutions for Differential Equations | 4th Edition

ISBN: 9780321964670

Solutions for Chapter 6.3: The Kernel and Range of a Linear Transformation

Solutions for Chapter 6.3
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Textbook: Differential Equations
Edition: 4
Author: Stephen W. Goode
ISBN: 9780321964670

Chapter 6.3: The Kernel and Range of a Linear Transformation includes 24 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. Since 24 problems in chapter 6.3: The Kernel and Range of a Linear Transformation have been answered, more than 20098 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

Key Math Terms and definitions covered in this textbook
  • 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!

  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Covariance matrix:E.

    When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

  • 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

  • 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.

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Hankel matrix H.

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

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Indefinite matrix.

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

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • 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.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • 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.

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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