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Solutions for Chapter 5: Linear Algebra with Applications 5th Edition

Linear Algebra with Applications | 5th Edition | ISBN: 9780321796974 | Authors: Otto Bretscher

Full solutions for Linear Algebra with Applications | 5th Edition

ISBN: 9780321796974

Linear Algebra with Applications | 5th Edition | ISBN: 9780321796974 | Authors: Otto Bretscher

Solutions for Chapter 5

Solutions for Chapter 5
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Textbook: Linear Algebra with Applications
Edition: 5
Author: Otto Bretscher
ISBN: 9780321796974

Chapter 5 includes 50 full step-by-step solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321796974. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 5. Since 50 problems in chapter 5 have been answered, more than 2339 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Fast Fourier Transform (FFT).

    A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

  • 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.

  • Gram-Schmidt orthogonalization A = QR.

    Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • 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.

  • 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.

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • Nullspace matrix N.

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

  • Permutation matrix P.

    There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Skew-symmetric matrix K.

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

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • Sum V + W of subs paces.

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

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Symmetric matrix A.

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

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

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