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Solutions for Chapter 11-2: Rational Expressions

Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition | ISBN: 9780078738227 | Authors: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more

Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition

ISBN: 9780078738227

Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition | ISBN: 9780078738227 | Authors: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more

Solutions for Chapter 11-2: Rational Expressions

Solutions for Chapter 11-2
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Textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1)
Edition: 1
Author: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more
ISBN: 9780078738227

Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. Chapter 11-2: Rational Expressions includes 67 full step-by-step solutions. Since 67 problems in chapter 11-2: Rational Expressions have been answered, more than 35276 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1.

Key Math Terms and definitions covered in this textbook
  • 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!

  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • Cholesky factorization

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

  • 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

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • 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

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • 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 .

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • Krylov subspace Kj(A, b).

    The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

  • Length II x II.

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

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Outer product uv T

    = column times row = rank one matrix.

  • Pivot.

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

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • 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 ().

  • 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.

  • Standard basis for Rn.

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

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