×
Log in to StudySoup
Get Full Access to Mathematics Education - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Mathematics Education - Textbook Survival Guide

Solutions for Chapter 9.5: Geometry

A Survey of Mathematics with Applications | 9th Edition | ISBN:  9780321759665 | Authors: Allen R. Angel, Christine D. Abbott, Dennis C. Runde

Full solutions for A Survey of Mathematics with Applications | 9th Edition

ISBN: 9780321759665

A Survey of Mathematics with Applications | 9th Edition | ISBN:  9780321759665 | Authors: Allen R. Angel, Christine D. Abbott, Dennis C. Runde

Solutions for Chapter 9.5: Geometry

Solutions for Chapter 9.5
4 5 0 259 Reviews
25
1
Textbook: A Survey of Mathematics with Applications
Edition: 9
Author: Allen R. Angel, Christine D. Abbott, Dennis C. Runde
ISBN: 9780321759665

Chapter 9.5: Geometry includes 64 full step-by-step solutions. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 64 problems in chapter 9.5: Geometry have been answered, more than 70930 students have viewed full step-by-step solutions from this chapter.

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).

  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

  • Cholesky factorization

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

  • Circulant matrix C.

    Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (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).

  • 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

  • Left inverse A+.

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

  • Length II x II.

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

  • 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.

  • 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.

  • Orthogonal subspaces.

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

  • Plane (or hyperplane) in Rn.

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

  • Row space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • Special solutions to As = O.

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

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • 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.

×
Log in to StudySoup
Get Full Access to Mathematics Education - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Mathematics Education - Textbook Survival Guide
×
Reset your password