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# Solutions for Chapter 5.4: Adding and Subtracting Polynomials; Graphing Simple Polynomials ## Full solutions for Beginning Algebra | 11th Edition

ISBN: 9780321673480 Solutions for Chapter 5.4: Adding and Subtracting Polynomials; Graphing Simple Polynomials

Solutions for Chapter 5.4
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##### ISBN: 9780321673480

This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. Since 108 problems in chapter 5.4: Adding and Subtracting Polynomials; Graphing Simple Polynomials have been answered, more than 37795 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.4: Adding and Subtracting Polynomials; Graphing Simple Polynomials includes 108 full step-by-step solutions. Beginning Algebra was written by and is associated to the ISBN: 9780321673480.

Key Math Terms and definitions covered in this textbook
• Cholesky factorization

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

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

• Cyclic shift

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

• Diagonal matrix D.

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

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• Eigenvalue A and eigenvector x.

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

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• 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).

• 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 ().

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• Spanning set.

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

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

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