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Solutions for Chapter 1.1: Systems of Linear Equations

Elementary Linear Algebra with Applications | 9th Edition | ISBN: 9780132296540 | Authors: Bernard Kolman David Hill

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

ISBN: 9780132296540

Elementary Linear Algebra with Applications | 9th Edition | ISBN: 9780132296540 | Authors: Bernard Kolman David Hill

Solutions for Chapter 1.1: Systems of Linear Equations

Solutions for Chapter 1.1
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Textbook: Elementary Linear Algebra with Applications
Edition: 9
Author: Bernard Kolman David Hill
ISBN: 9780132296540

This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 40 problems in chapter 1.1: Systems of Linear Equations have been answered, more than 12554 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. Chapter 1.1: Systems of Linear Equations includes 40 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • 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).

  • Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

    Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

  • Fundamental Theorem.

    The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Jordan form 1 = M- 1 AM.

    If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • 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 •

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

    Column vectors by convention.

  • 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.

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Symmetric matrix A.

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

  • Vector v in Rn.

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

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