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Solutions for Chapter 3: Parallel and Perpendicular Lines

Full solutions for Geometry | 1st Edition

ISBN: 9780030923456

Solutions for Chapter 3: Parallel and Perpendicular Lines

Solutions for Chapter 3
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Textbook: Geometry
Edition: 1
Author: Rinehart, Winston Holt
ISBN: 9780030923456

Chapter 3: Parallel and Perpendicular Lines includes 25 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Geometry, edition: 1. Geometry was written by and is associated to the ISBN: 9780030923456. Since 25 problems in chapter 3: Parallel and Perpendicular Lines have been answered, more than 47218 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Ellipse (or ellipsoid) x T Ax = 1.

    A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • 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.

  • Hankel matrix H.

    Constant along each antidiagonal; hij depends on i + j.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

  • 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.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)ยท(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 .

  • Nullspace matrix N.

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

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • Partial pivoting.

    In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

  • 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.

  • 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.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Standard basis for Rn.

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

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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