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# Solutions for Chapter 108: Transformations

## Full solutions for Saxon Math, Course 1 | 1st Edition

ISBN: 9781591417835

Solutions for Chapter 108: Transformations

Solutions for Chapter 108
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##### ISBN: 9781591417835

Chapter 108: Transformations includes 30 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 30 problems in chapter 108: Transformations have been answered, more than 33923 students have viewed full step-by-step solutions from this chapter. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1.

Key Math Terms and definitions covered in this textbook
• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Cayley-Hamilton Theorem.

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

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Diagonal matrix D.

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

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

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

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• 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].

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• 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.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• 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.

• Schwarz inequality

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

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Spectral Theorem A = QAQT.

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

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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