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Solutions for Chapter 4-4: Absolute Value Functions

Amsco's Algebra 2 and Trigonometry | 1st Edition | ISBN: 9781567657029 | Authors: Gantert

Full solutions for Amsco's Algebra 2 and Trigonometry | 1st Edition

ISBN: 9781567657029

Amsco's Algebra 2 and Trigonometry | 1st Edition | ISBN: 9781567657029 | Authors: Gantert

Solutions for Chapter 4-4: Absolute Value Functions

Solutions for Chapter 4-4
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Textbook: Amsco's Algebra 2 and Trigonometry
Edition: 1
Author: Gantert
ISBN: 9781567657029

Since 18 problems in chapter 4-4: Absolute Value Functions have been answered, more than 29458 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. Chapter 4-4: Absolute Value Functions includes 18 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Cayley-Hamilton Theorem.

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

  • 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.

  • 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).

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • Dimension of vector space

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

  • 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.

  • Free variable Xi.

    Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • 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.

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • 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.

  • Nullspace N (A)

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

  • 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.

  • Right inverse A+.

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

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

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

    Spectral radius = max of IAi I.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • 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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