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Solutions for Chapter 10.20: Warps and Morphs

Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition

ISBN: 9780470432051

Solutions for Chapter 10.20: Warps and Morphs

Chapter 10.20: Warps and Morphs includes 10 full step-by-step solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. This expansive textbook survival guide covers the following chapters and their solutions. Since 10 problems in chapter 10.20: Warps and Morphs have been answered, more than 15434 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10.

Key Math Terms and definitions covered in this textbook
  • Back substitution.

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

  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • 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.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • Hankel matrix H.

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

  • Norm

    IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

  • Nullspace matrix N.

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

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • 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.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Subspace S of V.

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

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

  • Symmetric matrix A.

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

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

  • Vector v in Rn.

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

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