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Solutions for Chapter 2.2: The Multiplication Property of Equality

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 2.2: The Multiplication Property of Equality

Solutions for Chapter 2.2
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Textbook: Introductory & Intermediate Algebra for College Students
Edition: 4
Author: Robert F. Blitzer
ISBN: 9780321758941

Since 114 problems in chapter 2.2: The Multiplication Property of Equality have been answered, more than 67401 students have viewed full step-by-step solutions from this chapter. Chapter 2.2: The Multiplication Property of Equality includes 114 full step-by-step solutions. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • Covariance matrix:E.

    When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • 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.

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • 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.

  • 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.

  • Identity matrix I (or In).

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

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • 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).

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Similar matrices A and B.

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

  • Singular Value Decomposition

    (SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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