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# Solutions for Chapter 11.7: Algebra and Trigonometry 9th Edition

## Full solutions for Algebra and Trigonometry | 9th Edition

ISBN: 9780321716569

Solutions for Chapter 11.7

Solutions for Chapter 11.7
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##### ISBN: 9780321716569

Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11.7 includes 64 full step-by-step solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Since 64 problems in chapter 11.7 have been answered, more than 55819 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.

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

The n roots are the eigenvalues of A.

• 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 IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

• 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

• Diagonal matrix D.

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

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Fourier matrix F.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

• Full column rank r = n.

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

• 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.

• Length II x II.

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

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• 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).

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• 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.

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