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# Solutions for Chapter 2.2: An Introduction to Problem Solving

## Full solutions for Intermediate Algebra | 6th Edition

ISBN: 9780321785046

Solutions for Chapter 2.2: An Introduction to Problem Solving

Solutions for Chapter 2.2
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##### ISBN: 9780321785046

Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.2: An Introduction to Problem Solving includes 86 full step-by-step solutions. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Since 86 problems in chapter 2.2: An Introduction to Problem Solving have been answered, more than 62471 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Cayley-Hamilton Theorem.

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

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of 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).

• 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.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• 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.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Particular solution x p.

Any solution to Ax = b; often x p has free variables = o.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• 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).

• 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 ().

• Similar matrices A and B.

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

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Trace of A

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

• Transpose matrix AT.

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

• Vector v in Rn.

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

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