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Solutions for Chapter 6.2: Inconsistent and Dependent Systems and Their Applications

College Algebra | 7th Edition | ISBN: 9780134469164 | Authors: Robert F. Blitzer

Full solutions for College Algebra | 7th Edition

ISBN: 9780134469164

College Algebra | 7th Edition | ISBN: 9780134469164 | Authors: Robert F. Blitzer

Solutions for Chapter 6.2: Inconsistent and Dependent Systems and Their Applications

Solutions for Chapter 6.2
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Textbook: College Algebra
Edition: 7
Author: Robert F. Blitzer
ISBN: 9780134469164

Chapter 6.2: Inconsistent and Dependent Systems and Their Applications includes 52 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780134469164. Since 52 problems in chapter 6.2: Inconsistent and Dependent Systems and Their Applications have been answered, more than 29818 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra , edition: 7.

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).

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of 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.

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • 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

  • Diagonalizable matrix A.

    Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

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

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

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • 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.

  • Particular solution x p.

    Any solution to Ax = b; often x p has free variables = o.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

  • Vandermonde matrix V.

    V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

  • Vector space V.

    Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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