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Solutions for Chapter 79: Area of a Triangle

Saxon Math, Course 1 | 1st Edition | ISBN: 9781591417835 | Authors: Stephan Hake

Full solutions for Saxon Math, Course 1 | 1st Edition

ISBN: 9781591417835

Saxon Math, Course 1 | 1st Edition | ISBN: 9781591417835 | Authors: Stephan Hake

Solutions for Chapter 79: Area of a Triangle

Solutions for Chapter 79
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Textbook: Saxon Math, Course 1
Edition: 1
Author: Stephan Hake
ISBN: 9781591417835

Chapter 79: Area of a Triangle includes 30 full step-by-step solutions. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 30 problems in chapter 79: Area of a Triangle have been answered, more than 33967 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Cramer's Rule for Ax = b.

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

  • 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

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • Dimension of vector space

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

  • 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.

  • 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.

  • 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.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Multiplication Ax

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

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • 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 •

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Similar matrices A and B.

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

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • 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.

  • Subspace S of V.

    Any vector space inside V, including V and Z = {zero vector only}.

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