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Solutions for Chapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

Full solutions for College Algebra | 9th Edition

ISBN: 9780321716811

Solutions for Chapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry

Solutions for Chapter 2.2
4 5 0 310 Reviews
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Textbook: College Algebra
Edition: 9
Author: Michael Sullivan
ISBN: 9780321716811

Chapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry includes 94 full step-by-step solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 9. College Algebra was written by and is associated to the ISBN: 9780321716811. This expansive textbook survival guide covers the following chapters and their solutions. Since 94 problems in chapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry have been answered, more than 32211 students have viewed full step-by-step solutions from this chapter.

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.

  • Cofactor Cij.

    Remove row i and column j; multiply the determinant by (-I)i + j •

  • Dimension of vector space

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

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • Inverse matrix A-I.

    Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Krylov subspace Kj(A, b).

    The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

  • 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.

  • Left inverse A+.

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

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Multiplication Ax

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

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

  • Plane (or hyperplane) in Rn.

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

  • Rank r (A)

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

  • Saddle point of I(x}, ... ,xn ).

    A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Toeplitz matrix.

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

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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