Solutions for Chapter 1.3: Graphing Equations

Full solutions for Intermediate Algebra for College Students | 6th Edition

ISBN: 9780321758934

Solutions for Chapter 1.3: Graphing Equations

Solutions for Chapter 1.3
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Textbook: Intermediate Algebra for College Students
Edition: 6
Author: Robert F. Blitzer
ISBN: 9780321758934

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Since 88 problems in chapter 1.3: Graphing Equations have been answered, more than 15991 students have viewed full step-by-step solutions from this chapter. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Chapter 1.3: Graphing Equations includes 88 full step-by-step solutions.

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

    Parentheses can be removed to leave ABC.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Complete solution x = x p + Xn to Ax = b.

    (Particular x p) + (x n in nullspace).

  • Covariance matrix:E.

    When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

  • Determinant IAI = det(A).

    Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Elimination.

    A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

  • 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.

  • 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.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • Identity matrix I (or In).

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

  • Linearly dependent VI, ... , Vn.

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

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • Pivot.

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

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

    Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

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