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Solutions for Chapter 3.1: Definition and Examples

Full solutions for Linear Algebra with Applications | 9th Edition

ISBN: 9780321962218

Solutions for Chapter 3.1: Definition and Examples

Since 16 problems in chapter 3.1: Definition and Examples have been answered, more than 10303 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. Chapter 3.1: Definition and Examples includes 16 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218.

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!

  • Circulant matrix C.

    Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

  • Complex conjugate

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

  • 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.

  • Elimination matrix = Elementary matrix Eij.

    The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • Fourier matrix F.

    Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Lucas numbers

    Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

  • Nullspace N (A)

    = All solutions to Ax = O. Dimension n - r = (# columns) - rank.

  • Orthonormal vectors q 1 , ... , q n·

    Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

  • 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 space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • 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!

  • Subspace S of V.

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

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

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