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Solutions for Chapter 1: Vectors

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition | ISBN: 9780538735452 | Authors: David Poole

Full solutions for Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition

ISBN: 9780538735452

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition | ISBN: 9780538735452 | Authors: David Poole

Solutions for Chapter 1: Vectors

Solutions for Chapter 1
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Textbook: Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign)
Edition: 3
Author: David Poole
ISBN: 9780538735452

Chapter 1: Vectors includes 214 full step-by-step solutions. Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) was written by and is associated to the ISBN: 9780538735452. Since 214 problems in chapter 1: Vectors have been answered, more than 12422 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign), edition: 3. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Cayley-Hamilton Theorem.

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

  • Cholesky factorization

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

  • 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.

  • 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.

  • 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].

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Nullspace N (A)

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

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Row picture of Ax = b.

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

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • Skew-symmetric matrix K.

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

  • 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}.

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

    T- 1 has rank 1 above and below diagonal.

  • Vandermonde matrix V.

    V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

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