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Solutions for Chapter 2: Chapter 2 Test

Introductory & Intermediate Algebra for College Students | 4th Edition | ISBN: 9780321758941 | Authors: Robert F. Blitzer

Full solutions for Introductory & Intermediate Algebra for College Students | 4th Edition

ISBN: 9780321758941

Introductory & Intermediate Algebra for College Students | 4th Edition | ISBN: 9780321758941 | Authors: Robert F. Blitzer

Solutions for Chapter 2: Chapter 2 Test

Solutions for Chapter 2
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Textbook: Introductory & Intermediate Algebra for College Students
Edition: 4
Author: Robert F. Blitzer
ISBN: 9780321758941

Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. Since 34 problems in chapter 2: Chapter 2 Test have been answered, more than 74816 students have viewed full step-by-step solutions from this chapter. Chapter 2: Chapter 2 Test includes 34 full step-by-step solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

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).

  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Complex conjugate

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

  • Cross product u xv in R3:

    Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

  • 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.

  • 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.

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Left nullspace N (AT).

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

  • Length II x II.

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

  • Nullspace N (A)

    = All solutions to Ax = O. Dimension n - r = (# columns) - rank.

  • 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.

  • Outer product uv T

    = column times row = rank one matrix.

  • Permutation matrix P.

    There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

  • 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).

  • Special solutions to As = O.

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

  • Subspace S of V.

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

  • Vector v in Rn.

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

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