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Solutions for Chapter 2: Parallel Lines

Elementary Geometry for College Students | 6th Edition | ISBN: 9781285195698 | Authors: Daniel C. Alexander, Geralyn M. Koeberlein

Full solutions for Elementary Geometry for College Students | 6th Edition

ISBN: 9781285195698

Elementary Geometry for College Students | 6th Edition | ISBN: 9781285195698 | Authors: Daniel C. Alexander, Geralyn M. Koeberlein

Solutions for Chapter 2: Parallel Lines

Solutions for Chapter 2
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Textbook: Elementary Geometry for College Students
Edition: 6
Author: Daniel C. Alexander, Geralyn M. Koeberlein
ISBN: 9781285195698

This textbook survival guide was created for the textbook: Elementary Geometry for College Students, edition: 6. Since 50 problems in chapter 2: Parallel Lines have been answered, more than 4221 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2: Parallel Lines includes 50 full step-by-step solutions. Elementary Geometry for College Students was written by and is associated to the ISBN: 9781285195698.

Key Math Terms and definitions covered in this textbook
  • Adjacency matrix of a graph.

    Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

  • Cramer's Rule for Ax = b.

    B j has b replacing column j of A; x j = det B j I det A

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Distributive Law

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

  • Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

    Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

  • 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!).

  • 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.

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • 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).

  • Orthogonal subspaces.

    Every v in V is orthogonal to every w in W.

  • Orthonormal vectors q 1 , ... , q n·

    Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Toeplitz matrix.

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

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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