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# Solutions for Chapter 4.3: FITTING LINEAR MODELS TO DATA

## Full solutions for College Algebra | 1st Edition

ISBN: 9781938168383

Solutions for Chapter 4.3: FITTING LINEAR MODELS TO DATA

Solutions for Chapter 4.3
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##### ISBN: 9781938168383

This textbook survival guide was created for the textbook: College Algebra, edition: 1. Chapter 4.3: FITTING LINEAR MODELS TO DATA includes 42 full step-by-step solutions. Since 42 problems in chapter 4.3: FITTING LINEAR MODELS TO DATA have been answered, more than 31900 students have viewed full step-by-step solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9781938168383. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Back substitution.

Upper triangular systems are solved in reverse order Xn to Xl.

• Big formula for n by n determinants.

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

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

The n roots are the eigenvalues of A.

• Cholesky factorization

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

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Hankel matrix H.

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

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Left nullspace N (AT).

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

• Nilpotent matrix N.

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

• Orthogonal subspaces.

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

• 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).

• 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.

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• 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 ().

• Solvable system Ax = b.

The right side b is in the column space of A.

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Wavelets Wjk(t).

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

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