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Solutions for Chapter 5.5: Orthogonal Complements

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

ISBN: 9780471669593

Solutions for Chapter 5.5: Orthogonal Complements

Solutions for Chapter 5.5
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Textbook: Elementary Linear Algebra with Applications
Edition: 9
Author: Howard Anton, Chris Rorres
ISBN: 9780471669593

Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 29 problems in chapter 5.5: Orthogonal Complements have been answered, more than 9126 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.5: Orthogonal Complements includes 29 full step-by-step solutions.

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

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

  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • Cramer's Rule for Ax = b.

    B j has b replacing column j of A; x j = det B j I det 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.

  • Dimension of vector space

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

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • 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).

  • Nilpotent matrix N.

    Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

  • 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.

  • 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.

  • 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 ].

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • Subspace S of V.

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

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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