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Solutions for Chapter 7.4: Adding, Subtracting, and Multiplying Radical Expressions

Intermediate Algebra | 6th Edition | ISBN: 9780321785046 | Authors: Elayn El Martin-Gay

Full solutions for Intermediate Algebra | 6th Edition

ISBN: 9780321785046

Intermediate Algebra | 6th Edition | ISBN: 9780321785046 | Authors: Elayn El Martin-Gay

Solutions for Chapter 7.4: Adding, Subtracting, and Multiplying Radical Expressions

Solutions for Chapter 7.4
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Textbook: Intermediate Algebra
Edition: 6
Author: Elayn El Martin-Gay
ISBN: 9780321785046

Since 92 problems in chapter 7.4: Adding, Subtracting, and Multiplying Radical Expressions have been answered, more than 59951 students have viewed full step-by-step solutions from this chapter. Chapter 7.4: Adding, Subtracting, and Multiplying Radical Expressions includes 92 full step-by-step solutions. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • 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.

  • 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!

  • 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.

  • Cofactor Cij.

    Remove row i and column j; multiply the determinant by (-I)i + j •

  • Complete solution x = x p + Xn to Ax = b.

    (Particular x p) + (x n in nullspace).

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • 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.

  • 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.

  • Identity matrix I (or In).

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

  • 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.

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • Plane (or hyperplane) in Rn.

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

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

    Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Spanning set.

    Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • 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.

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