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# Solutions for Chapter 1: College Algebra and Trigonometry 7th Edition

## Full solutions for College Algebra and Trigonometry | 7th Edition

ISBN: 9781439048603

Solutions for Chapter 1

Solutions for Chapter 1
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##### ISBN: 9781439048603

Since 85 problems in chapter 1 have been answered, more than 5920 students have viewed full step-by-step solutions from this chapter. Chapter 1 includes 85 full step-by-step solutions. College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

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

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

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

• Condition number

cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Orthogonal subspaces.

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

• Outer product uv T

= column times row = rank one matrix.

• 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).

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Pivot.

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

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

• 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).

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

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

Spectral radius = max of IAi I.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

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