Solutions for Chapter 15: Graphs, Charts, and Numbers

Full solutions for Excursions in Modern Mathematics | 8th Edition

ISBN: 9781292022048

Solutions for Chapter 15: Graphs, Charts, and Numbers

Solutions for Chapter 15
4 5 0 364 Reviews
28
2
Textbook: Excursions in Modern Mathematics
Edition: 8
Author: Peter Tannenbaum
ISBN: 9781292022048

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Excursions in Modern Mathematics, edition: 8. Chapter 15: Graphs, Charts, and Numbers includes 80 full step-by-step solutions. Since 80 problems in chapter 15: Graphs, Charts, and Numbers have been answered, more than 5073 students have viewed full step-by-step solutions from this chapter. Excursions in Modern Mathematics was written by and is associated to the ISBN: 9781292022048.

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).

  • Basis for V.

    Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

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

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

  • Elimination matrix = Elementary matrix Eij.

    The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

  • 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.

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • 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.

  • Hankel matrix H.

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

  • Kronecker product (tensor product) A ® B.

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

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • Linearly dependent VI, ... , Vn.

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

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • 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.

  • Pivot.

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

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

×
Log in to StudySoup
Get Full Access to Thousands of Study Materials at Your School

Forgot password? Reset password here

Join StudySoup for FREE
Get Full Access to Thousands of Study Materials at Your School
Join with Email
Already have an account? Login here
Reset your password

I don't want to reset my password

Need help? Contact support

Need an Account? Is not associated with an account
Sign up
We're here to help

Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or support@studysoup.com

Got it, thanks!
Password Reset Request Sent An email has been sent to the email address associated to your account. Follow the link in the email to reset your password. If you're having trouble finding our email please check your spam folder
Got it, thanks!
Already have an Account? Is already in use
Log in
Incorrect Password The password used to log in with this account is incorrect
Try Again

Forgot password? Reset it here